Mathematics: Number Sense and Operations – Grade 3

Numbers can be expressed in three different forms, just like you can describe your favorite toy in different ways! Let's explore how the same number can look completely different depending on how we write it.

Standard form is the way you're most familiar with seeing numbers. It uses digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) to show the exact value. For example, when you see 2,530, this is standard form. The comma helps us read larger numbers more easily by separating the thousands from the hundreds.

When we write numbers in standard form up to 10,000, we use these place values from right to left:

• Ones place (rightmost digit)
• Tens place (second digit from right)
• Hundreds place (third digit from right)
• Thousands place (leftmost digit)

Expanded form is like taking apart a toy to see all its pieces! 🧩 We break the number into parts based on the value of each digit's position. Let's look at our example 2,530:

• The 2 is in the thousands place, so it represents 2,000
• The 5 is in the hundreds place, so it represents 500
• The 3 is in the tens place, so it represents 30
• The 0 is in the ones place, so it represents 0

In expanded form, we write: 2,000 + 500 + 30

Notice how we don't include the + 0 because adding zero doesn't change the value! When a digit is zero, it means there are zero of that place value unit.

Word form writes out numbers using words instead of digits, just like when you write out your age in a story! For 2,530, we write two thousand five hundred thirty.

Here's how to read large numbers in word form:

1. Start from the left (biggest place value)
2. Read the thousands first: "two thousand"
3. Then read the hundreds: "five hundred"
4. Finally read the tens and ones together: "thirty"
5. Put it all together: "two thousand five hundred thirty"

Zero is a special digit that acts like a placeholder! 🎯 When zero appears in a number, it tells us there are zero of that place value unit. For example:

• In 1,008: there are 0 hundreds and 0 tens
• In 2,530: there are 0 ones
• In 5,000: there are 0 hundreds, 0 tens, and 0 ones

Zero is important because it holds the place so other digits stay in their correct positions. Without zero, 2,530 would become 253, which is a completely different number!

Let's practice with the number 3,079:

• Standard form: 3,079
• Expanded form: 3,000 + 70 + 9
• Word form: three thousand seventy-nine

Notice there are 0 hundreds in this number, so we don't include 0 in expanded form, and we skip "hundred" in word form.

Numbers in different forms appear everywhere in your daily life! 📚 You might see:

• Standard form: On price tags ($\$2,530$)
• Expanded form: When learning about money ($\$2,000 + \$500 + \$30$)
• Word form: When writing checks or in stories ("two thousand five hundred thirty dollars")

Understanding these different forms helps you become flexible with numbers and prepares you for more advanced math concepts!

Imagine numbers as flexible building blocks that you can take apart and put back together in many different ways! 🧱 Just like you can build the same tower using different combinations of blocks, you can represent the same number using different combinations of thousands, hundreds, tens, and ones.

Composing means putting parts together to make a whole number. Decomposing means taking a whole number apart into different parts. The amazing thing is that the same number can be decomposed in many different ways while still keeping the same total value!

Let's start with the number 5,783 and decompose it in the traditional way:

• 5 thousands + 7 hundreds + 8 tens + 3 ones
• This equals: 5,000 + 700 + 80 + 3 = 5,783

This is the most straightforward way to break apart a number, where each digit stays in its original place value.

Here's where it gets exciting! 🎉 We can regroup place values to create different representations of the same number. Remember these important relationships:

• 1 thousand = 10 hundreds
• 1 hundred = 10 tens
• 1 ten = 10 ones

Using 5,783 as our example, here are different ways to decompose it:

Method 1: Using only hundreds and ones

• Think: 5 thousands = 50 hundreds, plus the original 7 hundreds = 57 hundreds
• 8 tens = 80 ones, plus the original 3 ones = 83 ones
• Result: 57 hundreds + 83 ones = 5,700 + 83 = 5,783

Method 2: Using thousands and hundreds

• 8 tens and 3 ones = 83 ones, but we can regroup 80 ones as 8 tens
• We can also think of 8 tens as part of hundreds: 83 ones becomes 0.83 hundreds
• Actually, let's be more practical: 783 ones, or 7 hundreds + 83 ones
• Result: 5 thousands + 7 hundreds + 83 ones

Method 3: Using only tens and ones

• 5 thousands = 500 tens, 7 hundreds = 70 tens
• Total tens: 500 + 70 + 8 = 578 tens
• Result: 578 tens + 3 ones = 5,780 + 3 = 5,783

Visual models help us see these relationships clearly! Here are some tools:

Base-Ten Blocks 📦

• Large cubes represent thousands
• Flat squares represent hundreds
• Long rods represent tens
• Small cubes represent ones

You can physically trade blocks: 1 large cube for 10 flat squares, or 1 flat square for 10 long rods!

Place Value Charts

Each row shows the same number (5,783) decomposed differently!

Flexible decomposition helps you:

1. Understand addition and subtraction algorithms - When you need to "borrow" or "carry"
2. Make mental math easier - Breaking numbers into friendlier parts
3. Solve problems creatively - Finding the method that works best for you
4. Build number sense - Seeing relationships between place values

Imagine you're organizing a school supply collection! 📝 You have 5,783 pencils:

• Standard organization: 5 boxes of 1,000 + 7 boxes of 100 + 8 packs of 10 + 3 individual pencils
• Flexible organization: 57 boxes of 100 + 83 individual pencils
• Another way: 578 packs of 10 + 3 individual pencils

All represent the same total, but different organizations might work better for different situations!

Flexible decomposition is especially helpful in subtraction. For 5,783 - 892:

Instead of the traditional way, we might regroup:

• Original: 5 thousands + 7 hundreds + 8 tens + 3 ones
• Regroup for easier subtraction: 5 thousands + 6 hundreds + 18 tens + 3 ones
• Now we can subtract 9 tens from 18 tens easily!

This flexibility makes you a more confident and capable mathematician! 🌟

Numbers have relationships just like people do! Some numbers are bigger, some are smaller, and some are exactly equal. Learning to compare numbers is like being a detective 🕵️‍♀️, looking for clues in each digit's place value to determine which number wins the "bigger number" contest!

When we compare two numbers, there are only three possible relationships:

• Greater than (>) - The first number is bigger
• Less than (<) - The first number is smaller
• Equal to (=) - Both numbers are exactly the same

We also use not equal to (≠) when numbers are different, but we haven't determined which is larger yet.

The secret to comparing numbers is to start from the leftmost digit (greatest place value) and work your way right, just like reading a book! 📖

Let's compare 6,909 and 6,099:

Step 1: Compare thousands place

• 6,909 has 6 thousands
• 6,099 has 6 thousands
• They're equal, so move to the next place!

Step 2: Compare hundreds place

• 6,909 has 9 hundreds (represents 900)
• 6,099 has 0 hundreds (represents 0)
• Since 9 > 0, we know 6,909 > 6,099!

We don't need to look at tens or ones because the hundreds place already tells us the answer.

Number lines are like rulers for numbers! 📏 They help us see which numbers are closer to zero and which are farther away.

0 -------- 3,475 -------- 4,743 -- 4,753 -------- 10,000

On this number line:

• 3,475 is closest to zero, so it's the smallest
• 4,753 is farthest from zero, so it's the largest
• 4,743 and 4,753 are very close to each other!

Number lines can be scaled by different amounts (50s, 100s, 1,000s) depending on the range of numbers we're working with.

Example: Plotting 3,790; 3,890; 3,799; 3,809

On a number line from 3,700 to 3,900 (scaled by 50s):

3,700 -- 3,750 -- 3,790 -- 3,799 -- 3,809 -- 3,850 -- 3,890 -- 3,900

Notice how 3,799 and 3,809 are very close together, while 3,890 is much farther to the right.

It's important to read the entire comparison statement, not just the symbol! Practice reading these both ways:

• 7,309 > 7,039 reads "7,309 is greater than 7,039"
• 7,039 < 7,309 reads "7,039 is less than 7,309"

Both statements are true and mean the same thing!

Scenario 1: Different number of digits

• 999 vs 1,000
• Any 4-digit number is automatically greater than any 3-digit number!

Scenario 2: Same digits, different order

• 3,475 vs 3,547
• Compare place by place: thousands equal (3), hundreds equal (4 vs 5)
• Since 5 > 7 is false, actually 4 < 5, so 3,475 < 3,547

Scenario 3: Numbers with zeros

• 5,007 vs 5,070
• Thousands equal (5), hundreds equal (0), tens: 0 vs 7
• Since 0 < 7, we have 5,007 < 5,070

When putting several numbers in order, compare them two at a time:

Numbers to order: 8,403; 8,340; 8,034; 8,304

1. All have 8 thousands, so compare hundreds:

   • 8,403 has 4 hundreds
   • 8,340 has 3 hundreds
   • 8,034 has 0 hundreds
   • 8,304 has 3 hundreds

2. Start with smallest hundreds (0): 8,034 is smallest

3. Next are the 3 hundreds: 8,340 vs 8,304

   • Compare tens: 4 vs 0, so 8,340 > 8,304
   • Order so far: 8,034, 8,304, 8,340

4. Finally 8,403 (4 hundreds) is largest

Final order: 8,034 < 8,304 < 8,340 < 8,403

Visual models help make comparisons concrete! 🧮 When comparing 274 and 312 with base-ten blocks:

• 274: 2 hundreds flats + 7 tens rods + 4 ones cubes
• 312: 3 hundreds flats + 1 tens rod + 2 ones cubes

You can immediately see that 312 has more hundreds flats, so 312 > 274!

Comparing numbers happens everywhere! 🌍

• Sports scores: Which team won? 84 > 76
• Shopping: Which costs more? $\$3,250$ > $\$2,850$
• Distances: Which city is farther? 1,247 miles > 987 miles
• Population: Which school is bigger? 2,156 students > 1,987 students

Mastering number comparison helps you make sense of data and solve problems in everyday life!

Rounding is like finding the closest landmark when giving directions! 🗺️ Instead of saying "meet me at house number 347," you might say "meet me near 350" because it's easier to remember and find. Rounding helps us estimate and work with numbers that are easier to calculate with, while still being close to the original number.

Rounding means finding the nearest benchmark number that's easier to work with. When we round to the nearest 10, we're looking for the closest number that ends in 0. When we round to the nearest 100, we're looking for the closest number that ends in 00.

Benchmark numbers are like signposts on a number line! 🚩 They help us know where we are and where we're going.

For rounding to the nearest 10:

• Benchmark tens: ...630, 640, 650, 660, 670...
• These are numbers that end in 0

For rounding to the nearest 100:

• Benchmark hundreds: ...600, 700, 800, 900, 1000...
• These are numbers that end in 00

Let's round 643 to the nearest ten using a number line:

Step 1: Identify the benchmark tens

• 643 falls between 640 and 650
• These are our two choices!

Step 2: Find the halfway point

• Halfway between 640 and 650 is 645
• Numbers from 640-644 are closer to 640
• Numbers from 646-650 are closer to 650
• The number 645 is exactly in the middle

Step 3: Determine which is closer

• 643 is only 3 away from 640 (643 - 640 = 3)
• 643 is 7 away from 650 (650 - 643 = 7)
• Since 3 < 7, we round down to 640

640 -------- 643 ---- 645 -------- 650
    3 away          2 away from middle

Let's round 439 to the nearest hundred:

Step 1: Identify benchmark hundreds

• 439 falls between 400 and 500

Step 2: Find the halfway point

• Halfway between 400 and 500 is 450

Step 3: Determine which is closer

• 439 is 39 away from 400 (439 - 400 = 39)
• 439 is 61 away from 500 (500 - 439 = 61)
• Since 39 < 61, we round down to 400

400 ---------- 439 -- 450 ---------- 500
    39 away         11 away from middle

What happens when a number is exactly halfway between two benchmarks? 🤔

Example: Rounding 75 to the nearest ten

• 75 is exactly halfway between 70 and 80
• The special rule says: always round up when exactly halfway
• So 75 rounds to 80

Example: Rounding 450 to the nearest hundred

• 450 is exactly halfway between 400 and 500
• Following the rule: 450 rounds to 500

Rounding helps us in many practical ways:

1. Estimation: Before calculating 347 + 289, round to 350 + 290 = 640 to estimate
2. Mental math: It's easier to work with 600 + 400 than 617 + 383
3. Real-world communication: "About 500 people" is clearer than "487 people"
4. Checking answers: If 347 + 289 = 636, and our estimate was 640, that seems reasonable!

For any rounding problem:

1. Identify what you're rounding to (tens or hundreds)
2. Find the two benchmarks your number falls between
3. Locate the halfway point between those benchmarks
4. Determine which benchmark is closer to your number
5. Apply the halfway rule if needed (round up when exactly halfway)

Example 1: Round 276 to the nearest ten

• Benchmarks: 270 and 280
• Halfway point: 275
• 276 is 1 away from 275, so it's closer to 280
• Answer: 280

Example 2: Round 850 to the nearest hundred

• Benchmarks: 800 and 900
• Halfway point: 850
• Since 850 is exactly halfway, round up
• Answer: 900

Example 3: Round 234 to the nearest hundred

• Benchmarks: 200 and 300
• Halfway point: 250
• 234 is closer to 200 (34 away vs 66 away)
• Answer: 200

❌ Wrong thinking: "Round 923 to the nearest ten... I see 2 tens, so round to 900" ✅ Correct thinking: "Look at the ones digit (3) to help determine if 923 is closer to 920 or 930"

❌ Wrong thinking: "920 can't be rounded because it's already at a benchmark" ✅ Correct thinking: "920 rounded to the nearest ten is still 920"

Rounding appears everywhere in daily life! 💰

• Money: "That costs about $\$30$" (instead of $\$27.83$)
• Time: "I'll be there in about 20 minutes" (instead of 17 minutes)
• Attendance: "About 500 people came" (instead of 487)
• Distances: "It's about 300 miles" (instead of 284 miles)

Mastering rounding gives you a powerful tool for estimation and helps you determine if your calculations make sense!

Becoming fluent with addition and subtraction algorithms is like learning to ride a bike – once you master the steps and understand why they work, you'll be able to calculate quickly and confidently! 🚴‍♀️ Let's explore how to add and subtract large numbers using strategies that mathematicians have been using for centuries.

A standard algorithm is a step-by-step procedure that always works for solving a particular type of problem. Think of it as a recipe that you can follow to get the right answer every time! The most important thing is that you understand WHY each step works, not just HOW to do it.

Let's work through 174 + 253 step by step:

Step 1: Estimate First 📊 Before calculating, let's estimate to make sure our answer will be reasonable:

• 174 is close to 200
• 253 is close to 250
• 200 + 250 = 450, so our answer should be close to 450

Step 2: Set Up Vertically

  1 7 4
+ 2 5 3
-------

Step 3: Add from Right to Left (Ones Place)

• 4 ones + 3 ones = 7 ones
• Write 7 in the ones place
• No regrouping needed since 7 < 10

Step 4: Add Tens Place

• 7 tens + 5 tens = 12 tens
• 12 tens = 1 hundred + 2 tens
• Write 2 in the tens place, carry 1 hundred to hundreds place

Step 5: Add Hundreds Place

• 1 hundred + 2 hundreds + 1 hundred (carried) = 4 hundreds
• Write 4 in the hundreds place

Final Answer: 427

Our answer of 427 is close to our estimate of 450, so it's reasonable! ✅

Regrouping happens when we have 10 or more of any place value unit. Let's see why this works:

Example: 7 tens + 5 tens = 12 tens

• We can't write "12" in the tens place
• Instead, we regroup: 12 tens = 10 tens + 2 tens
• 10 tens = 1 hundred
• So we write 2 tens and carry 1 hundred

This is just like trading money: 12 dimes = 1 dollar + 2 dimes! 💰

Let's work through 327 - 174:

Step 1: Estimate First

• 327 is close to 330
• 174 is close to 170
• 330 - 170 = 160, so our answer should be close to 160

Step 2: Set Up Vertically

  3 2 7
- 1 7 4
-------

Step 3: Subtract from Right to Left (Ones Place)

• 7 ones - 4 ones = 3 ones ✅

Step 4: Subtract Tens Place

• We need to subtract 7 tens from 2 tens
• But 2 < 7, so we need to regroup!
• Borrow 1 hundred from hundreds place: 3 hundreds becomes 2 hundreds
• 1 hundred = 10 tens, so 2 tens + 10 tens = 12 tens
• Now: 12 tens - 7 tens = 5 tens ✅

Step 5: Subtract Hundreds Place

• 2 hundreds - 1 hundred = 1 hundred ✅

Final Answer: 153

Our answer of 153 is close to our estimate of 160! ✅

Regrouping in subtraction is like making change! 🏪 When you don't have enough of a smaller unit, you "break" a larger unit:

Example: Need to subtract 7 tens from 2 tens

• "Break" 1 hundred into 10 tens
• Now you have 2 tens + 10 tens = 12 tens
• 12 tens - 7 tens = 5 tens

It's like trading a $\$10$ bill for ten $\$1$ bills when you need to make change!

Base-Ten Blocks 🧮

• Large cubes = thousands
• Flat squares = hundreds
• Long rods = tens
• Small cubes = ones

You can physically trade blocks: 1 flat square for 10 long rods when you need to regroup!

Place Value Charts

This shows the regrouping step visually!

Commutative Property: Order doesn't matter in addition

• 174 + 253 = 253 + 174 = 427

Associative Property: Grouping doesn't matter

• (100 + 200) + (74 + 53) = 300 + 127 = 427

While the standard algorithm is efficient, you can also use other strategies:

Mental Math Strategy for 174 + 253:

• 174 + 253
• = 174 + 200 + 53
• = 374 + 53
• = 427

Break Apart Strategy for 327 - 174:

• 327 - 174
• = 327 - 100 - 74
• = 227 - 74
• = 227 - 70 - 4
• = 157 - 4
• = 153

Example: Miranda finds 492 seashells during her vacation. She now has 1,045 seashells in her collection. How many seashells did she have before vacation?

This is a subtraction problem: 1,045 - 492 = ?

Step 1: Estimate

• About 1,000 - 500 = 500 seashells

Step 2: Calculate using algorithm

  1 0 4 5
-   4 9 2
---------
    5 5 3

Step 3: Check reasonableness

• 553 is close to our estimate of 500 ✅
• Check: 553 + 492 should equal 1,045 ✅

Miranda had 553 seashells before her vacation! 🐚

1. Always estimate first to check if your answer is reasonable
2. Line up place values carefully when writing problems vertically
3. Work from right to left (ones, then tens, then hundreds)
4. Understand regrouping as trading between place values
5. Practice explaining your steps to build deeper understanding
6. Use visual models when you get stuck

Mastering these algorithms gives you the confidence to tackle any addition or subtraction problem! 🌟

Welcome to the exciting world of multiplication and division! 🎉 These operations are like discovering mathematical shortcuts that help you solve problems much faster than counting one by one. Think of multiplication as "super addition" and division as "super subtraction" – they're powerful tools that will make you a more efficient problem solver!

Multiplication is a way to find the total when you have equal groups of objects. Instead of adding the same number over and over, multiplication gives you a shortcut!

Example: If you have 4 groups of 6 stickers each, instead of calculating: 6 + 6 + 6 + 6 = 24 stickers

You can multiply: 4 × 6 = 24 stickers! 🌟

Multiplication answers the question: "How many in total when I have ___ groups of ___ each?"

Arrays 📐 An array is a rectangular arrangement of objects in rows and columns:

⭐ ⭐ ⭐ ⭐ ⭐ ⭐
⭐ ⭐ ⭐ ⭐ ⭐ ⭐
⭐ ⭐ ⭐ ⭐ ⭐ ⭐
⭐ ⭐ ⭐ ⭐ ⭐ ⭐

This shows 4 rows of 6 stars = 4 × 6 = 24 stars Or 6 columns of 4 stars = 6 × 4 = 24 stars

Equal Groups 👥 Think about organizing objects into equal-sized groups:

• 3 tables with 4 students each = 3 × 4 = 12 students
• 5 bags with 7 marbles each = 5 × 7 = 35 marbles

Area Models 📦 Drawing rectangles where the area represents the product:

• A rectangle that is 8 units long and 5 units wide has an area of 8 × 5 = 40 square units

Division is the opposite of multiplication! It helps you break apart a total into equal groups or find how many groups you can make.

There are two types of division problems:

Sharing (Partitive) Division 🍕 "I have 24 pizza slices to share equally among 6 friends. How many slices does each friend get?"

• Total: 24 slices
• Number of groups: 6 friends
• Find: How many in each group?
• Answer: 24 ÷ 6 = 4 slices per friend

Grouping (Quotative) Division 📚 "I have 24 books to put into boxes. Each box holds 6 books. How many boxes do I need?"

• Total: 24 books
• Size of each group: 6 books
• Find: How many groups?
• Answer: 24 ÷ 6 = 4 boxes needed

Multiplication and division are inverse operations – they undo each other! This creates fact families:

If you know that 4 × 6 = 24, then you also know:

• 6 × 4 = 24 (commutative property)
• 24 ÷ 4 = 6 (division undoes multiplication)
• 24 ÷ 6 = 4 (division the other way)

These four facts form a fact family! 👨‍👩‍👧‍👦

Let's use an array to explore the fact family for 6 × 7:

🔵 🔵 🔵 🔵 🔵 🔵 🔵
🔵 🔵 🔵 🔵 🔵 🔵 🔵
🔵 🔵 🔵 🔵 🔵 🔵 🔵
🔵 🔵 🔵 🔵 🔵 🔵 🔵
🔵 🔵 🔵 🔵 🔵 🔵 🔵
🔵 🔵 🔵 🔵 🔵 🔵 🔵

From this array, we can see:

• 6 rows × 7 columns = 42 total dots
• 7 columns × 6 rows = 42 total dots
• 42 ÷ 6 = 7 dots per row
• 42 ÷ 7 = 6 rows

Multiplication Example: "Tina has 4 shelves on her bookshelf. Each shelf has 6 books. How many books are on Tina's bookshelf in all?"

Step 1: Identify the pattern

• 4 groups (shelves)
• 6 items in each group (books per shelf)
• This is a multiplication problem!

Step 2: Draw a model

Shelf 1: 📖 📖 📖 📖 📖 📖
Shelf 2: 📖 📖 📖 📖 📖 📖
Shelf 3: 📖 📖 📖 📖 📖 📖
Shelf 4: 📖 📖 📖 📖 📖 📖

Step 3: Write the equation 4 × 6 = 24 books

Division Example: "A total of 56 chairs are in the cafeteria for an assembly. The principal arranges the chairs into 8 rows with the same number of chairs in each row. How many chairs are in each row?"

Step 1: Identify the pattern

• Total: 56 chairs
• Number of groups: 8 rows
• Find: How many in each group? (sharing division)

Step 2: Use the division equation 56 ÷ 8 = 7 chairs per row

Step 3: Check using multiplication 8 × 7 = 56 ✅

Skip counting helps you see multiplication patterns! 🔄

Counting by 3s: 3, 6, 9, 12, 15, 18, 21, 24... This shows: 1×3=3, 2×3=6, 3×3=9, 4×3=12, etc.

Counting by 5s: 5, 10, 15, 20, 25, 30, 35, 40... This shows: 1×5=5, 2×5=10, 3×5=15, 4×5=20, etc.

The same problem can be shown in different ways:

Problem: 3 × 8 = ?

Equal Groups: 🎾🎾🎾🎾🎾🎾🎾🎾 🎾🎾🎾🎾🎾🎾🎾🎾 🎾🎾🎾🎾🎾🎾🎾🎾

Array:

🎾 🎾 🎾 🎾 🎾 🎾 🎾 🎾
🎾 🎾 🎾 🎾 🎾 🎾 🎾 🎾
🎾 🎾 🎾 🎾 🎾 🎾 🎾 🎾

Repeated Addition: 8 + 8 + 8 = 24

Number Line: Jump by 8s three times

0 ----8----16----24
  +8   +8   +8

All methods show that 3 × 8 = 24! 🎯

❌ Confusing the factors: In 3 × 8, mixing up "3 groups of 8" with "8 groups of 3" ✅ Clear thinking: Use context clues to determine which number represents groups and which represents the size of each group

❌ Forgetting division types: Not distinguishing between sharing and grouping ✅ Clear thinking: Read carefully to determine what the problem is asking you to find

While you're exploring these concepts, you're building toward knowing multiplication and division facts quickly and accurately. The key is understanding the WHY behind each fact so you can:

• Remember facts more easily
• Figure out unknown facts using known ones
• Solve problems flexibly
• Check if your answers make sense

Remember: Understanding comes first, then speed! 🚀

Get ready to discover some amazing multiplication patterns! 🎯 When you multiply by multiples of 10 and 100, you're not just calculating – you're exploring how place value works in powerful ways. These patterns will become the foundation for multiplying much larger numbers in the future!

Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90 Multiples of 100: 100, 200, 300, 400, 500, 600, 700, 800, 900

These numbers are special because they end in zeros, and this creates predictable patterns when we multiply!

The secret to multiplying by multiples of 10 and 100 is connecting them to multiplication facts you already know! Let's start with a fact you know and build from there.

Starting with 6 × 7 = 42

Now let's see what happens when we multiply by multiples:

• 6 × 7 = 42 (6 ones × 7 ones = 42 ones)
• 6 × 70 = 420 (6 ones × 7 tens = 42 tens = 420 ones)
• 60 × 7 = 420 (6 tens × 7 ones = 42 tens = 420 ones)
• 6 × 700 = 4,200 (6 ones × 7 hundreds = 42 hundreds = 4,200 ones)
• 600 × 7 = 4,200 (6 hundreds × 7 ones = 42 hundreds = 4,200 ones)

Do you see the pattern? We're using the same basic fact (6 × 7 = 42) but working with different place values! 🔍

Example: 3 × 40

Let's think about what 40 really means:

• 40 = 4 tens
• So 3 × 40 = 3 × (4 tens) = 3 groups of 4 tens = 12 tens
• 12 tens = 120 ones
• Therefore: 3 × 40 = 120

Example: 5 × 300

Let's think about what 300 really means:

• 300 = 3 hundreds
• So 5 × 300 = 5 × (3 hundreds) = 5 groups of 3 hundreds = 15 hundreds
• 15 hundreds = 1,500 ones
• Therefore: 5 × 300 = 1,500

Imagine using base-ten blocks to show these patterns! 🧮

For 3 × 40:

• 40 is represented by 4 ten-rods
• 3 × 40 means 3 groups of 4 ten-rods
• Total: 12 ten-rods = 120 unit cubes

For 5 × 300:

• 300 is represented by 3 hundred-flats
• 5 × 300 means 5 groups of 3 hundred-flats
• Total: 15 hundred-flats = 1,500 unit cubes

Number lines help us see these patterns clearly! 📏

For 4 × 20 (skip counting by 20s):

0 ----20----40----60----80
  +20  +20  +20  +20

Four jumps of 20 lands us at 80, so 4 × 20 = 80!

For 3 × 200 (skip counting by 200s):

0 ----200----400----600
   +200  +200  +200

Three jumps of 200 lands us at 600, so 3 × 200 = 600!

You might hear someone say "just add zeros to the end," but this doesn't help you understand WHY the pattern works! 🚫

The problem with the "zero trick":

• For 5 × 8 = 40, if you use "5 × 80, just add one zero" you'd get 400 (wrong!)
• But 5 × 80 = 400, and 5 × 8 = 40 already has a zero!
• This trick fails when the basic fact already ends in zero

The place value way is better:

• 5 × 80 = 5 × (8 tens) = 40 tens = 400 ✅
• Understanding place value always works!

Look at these patterns on a hundreds chart! 📊

Multiplying by 10:

• 1 × 10 = 10
• 2 × 10 = 20
• 3 × 10 = 30
• Pattern: The product is the first factor with a 0 added

Why this works:

• 3 × 10 = 3 × (1 ten) = 3 tens = 30 ones

Example: At Sunnyland Amusement Park, adult tickets cost $\$20$ each. How much does entry cost for 9 adults?

Step 1: Set up the problem

• 9 adults × $\$20$ per adult = ?

Step 2: Use place value thinking

• $\$20$ = 2 tens dollars
• 9 × 2 tens = 18 tens = $\$180$

Step 3: Check with skip counting $\$20$, $\$40$, $\$60$, $\$80$, $\$100$, $\$120$, $\$140$, $\$160$, $\$180$ ✅

Answer: 9 adult tickets cost $\$180$

Example: What is 7 × 600?

Step 1: Use place value reasoning

• 600 = 6 hundreds
• 7 × 6 hundreds = 42 hundreds
• 42 hundreds = 4,200 ones

Step 2: Check using known facts

• We know 7 × 6 = 42
• So 7 × 600 = 4,200 ✅

Example: Find two different equations using a one-digit number and a multiple of 10 that equal 120.

Think: What basic fact gives us 12?

• 3 × 4 = 12, so 3 × 40 = 120 ✅
• 4 × 3 = 12, so 4 × 30 = 120 ✅
• 6 × 2 = 12, so 6 × 20 = 120 ✅
• 12 × 1 = 12, so 12 × 10 = 120 ✅

These patterns are building blocks for bigger concepts! 🏗️

For multi-digit multiplication:

• 34 × 8 = (30 × 8) + (4 × 8) = 240 + 32 = 272
• You'll use your knowledge that 30 × 8 = 3 tens × 8 = 24 tens = 240

For place value understanding:

• Understanding that 40 = 4 tens helps with regrouping
• Seeing patterns helps with mental math strategies

1. Start with known facts (like 6 × 7 = 42)
2. Apply place value thinking (6 × 70 = 6 × 7 tens = 42 tens = 420)
3. Use visual models when you get stuck
4. Check with skip counting for smaller numbers
5. Connect to real-world problems to build meaning

Remember: Understanding the WHY behind these patterns makes you a stronger mathematician than just memorizing tricks! 🌟

Now it's time to build your multiplication and division fact toolkit! 🧰 Think of multiplication and division facts as essential tools that every mathematician needs – the better you understand these tools and can use them reliably, the more complex and interesting problems you'll be able to solve!

Procedural reliability means you can solve multiplication and division problems accurately and efficiently using methods that make sense to you. It's not about memorizing without understanding – it's about developing strategies you trust and can explain to others! 🎯

The goal is to find methods that work reliably for YOU, whether that's using arrays, skip counting, known facts, or other strategies.

In your earlier exploration of multiplication and division, you discovered many important concepts. Now we're building on that foundation to develop reliable methods for facts with factors up to 12.

Understanding multiplication properties gives you tools to make facts easier! 💪

Commutative Property: Order Doesn't Matter

• 4 × 6 = 6 × 4 = 24
• This means you only need to learn half as many facts!
• If you know 7 × 9, you automatically know 9 × 7

Visual proof with arrays:

4 × 6 array:           6 × 4 array:
⭐⭐⭐⭐⭐⭐           ⭐⭐⭐⭐
⭐⭐⭐⭐⭐⭐           ⭐⭐⭐⭐
⭐⭐⭐⭐⭐⭐           ⭐⭐⭐⭐
⭐⭐⭐⭐⭐⭐           ⭐⭐⭐⭐
                      ⭐⭐⭐⭐
                      ⭐⭐⭐⭐

Both arrays have 24 stars – just rotated! 🔄

Distributive Property: Breaking Apart Numbers

• 6 × 8 = 6 × (5 + 3) = (6 × 5) + (6 × 3) = 30 + 18 = 48
• Or: 6 × 8 = (4 + 2) × 8 = (4 × 8) + (2 × 8) = 32 + 16 = 48

This lets you use easier facts to find harder ones!

Strategy 1: Use Known Facts to Find Unknown Facts

If you know 5 × 6 = 30, you can find 6 × 6:

• 6 × 6 = (5 + 1) × 6 = (5 × 6) + (1 × 6) = 30 + 6 = 36

Strategy 2: Double and Near-Doubles

• 6 × 4 = double of 6 × 2 = double of 12 = 24
• 7 × 6 = (7 × 5) + (7 × 1) = 35 + 7 = 42

Strategy 3: Skip Counting Patterns

• For 8 × 7: 7, 14, 21, 28, 35, 42, 49, 56
• The 8th number in the "counting by 7s" pattern is 56

Strategy 4: Array Visualization For 9 × 6, imagine a 9 × 6 array:

⚫⚫⚫⚫⚫⚫
⚫⚫⚫⚫⚫⚫
⚫⚫⚫⚫⚫⚫
⚫⚫⚫⚫⚫⚫
⚫⚫⚫⚫⚫⚫
⚫⚫⚫⚫⚫⚫
⚫⚫⚫⚫⚫⚫
⚫⚫⚫⚫⚫⚫
⚫⚫⚫⚫⚫⚫

You can count or break it into easier parts: (5 × 6) + (4 × 6) = 30 + 24 = 54

Every multiplication fact gives you division facts for free! This is the power of inverse operations.

From 6 × 7 = 42, we get:

• 42 ÷ 6 = 7 ("42 divided into groups of 6 gives 7 groups")
• 42 ÷ 7 = 6 ("42 divided into groups of 7 gives 6 groups")

Visual connection using arrays: The same 6 × 7 array can show division:

🔴🔴🔴🔴🔴🔴🔴
🔴🔴🔴🔴🔴🔴🔴
🔴🔴🔴🔴🔴🔴🔴
🔴🔴🔴🔴🔴🔴🔴
🔴🔴🔴🔴🔴🔴🔴
🔴🔴🔴🔴🔴🔴🔴

• Multiplication view: 6 rows of 7 = 42 total
• Division view 1: 42 objects in 6 rows = 7 in each row
• Division view 2: 42 objects in 7 columns = 6 in each column

A great way to practice is using fact family triangles! 🔺

        42
       /  \
      /    \
     6  ×÷  7

From this triangle, you can make:

• 6 × 7 = 42
• 7 × 6 = 42
• 42 ÷ 6 = 7
• 42 ÷ 7 = 6

Multiplication Example: "Show how to find the product of 6 × 7 in two different ways."

Method 1: Skip counting 7, 14, 21, 28, 35, 42 Six jumps of 7 gets us to 42.

Method 2: Breaking apart 6 × 7 = 6 × (5 + 2) = (6 × 5) + (6 × 2) = 30 + 12 = 42

Division Example: "Provide two division facts that have a quotient of 8."

Think: "What times 8 equals...?"

• 8 × 8 = 64, so 64 ÷ 8 = 8 ✅
• 4 × 8 = 32, so 32 ÷ 4 = 8 ✅
• 9 × 8 = 72, so 72 ÷ 9 = 8 ✅

The goal isn't just to memorize facts – it's to understand them so well that you can:

• Recall them quickly when needed
• Figure out forgotten facts using strategies
• Check if answers make sense using related facts
• Explain your thinking to others
• Apply facts to solve problems in new situations

Reliability comes first, then speed! Here's how to build both:

1. Understand the concept using visual models and real-world contexts
2. Develop reliable strategies that work for you
3. Practice regularly with a focus on accuracy
4. Connect facts to each other using properties and patterns
5. Apply facts in problems to build meaning
6. Build speed gradually while maintaining understanding

Multiplying by 0: Always equals 0

• 7 × 0 = 0 (zero groups of 7 = 0)

Multiplying by 1: Equals the other factor

• 9 × 1 = 9 (one group of 9 = 9)

Multiplying by 2: Doubling

• 8 × 2 = 16 (double 8 = 16)

Multiplying by 5: Patterns with 5 and 0

• 5 × even number ends in 0
• 5 × odd number ends in 5

Multiplying by 9: Finger trick and patterns

• 9 × 4 = 36 (digits add to 9: 3 + 6 = 9)

🤔 Ask yourself:

• Can I solve this fact in multiple ways?
• Does my answer make sense?
• Can I explain WHY this fact is true?
• What related facts do I know?
• How can I use this fact to find other facts?

When you can answer these questions confidently, you've developed true procedural reliability! 🌟