Expanding number sense and problem-solving skills
When we add or subtract two-digit numbers, we line up the ones and tens carefully. **Addition with Regrouping:** When the ones column adds up to 10 or more, we regroup by carrying 1 ten to the tens column. **Subtraction with Regrouping:** When we can't subtract the ones digit (because it's smaller), we borrow 1 ten from the tens place and turn it into 10 ones. **The Standard Algorithm:** This is the method where we stack numbers vertically and work column by column, starting from the ones place.
Example 1: 47 + 36
Step 1: Line up the numbers by place value. Write 47 on top and 36 below it, with ones under ones and tens under tens.
Step 2: Add the ones: 7 + 6 = 13. That's more than 10!
Step 3: Write the 3 in the ones place and carry the 1 ten to the tens column.
Step 4: Add the tens: 4 + 3 + 1 (carried) = 8 tens.
Step 5: Final answer: 83
Example 2: 82 - 37
Step 1: Line up the numbers. Write 82 on top and 37 below it.
Step 2: Look at the ones: Can we subtract 7 from 2? No! We need to regroup.
Step 3: Borrow 1 ten from the 8 tens. Now we have 7 tens and 12 ones.
Step 4: Subtract the ones: 12 - 7 = 5
Step 5: Subtract the tens: 7 - 3 = 4
Step 6: Final answer: 45
Example 3: 55 + 28
Step 1: Stack the numbers with ones and tens aligned.
Step 2: Add the ones: 5 + 8 = 13
Step 3: Write 3 in the ones place, carry 1 to the tens.
Step 4: Add the tens: 5 + 2 + 1 = 8
Step 5: Final answer: 83
Question 1: What is 46 + 27?
Question 2: What is 91 - 48?
Question 3: What is 34 + 59?
Question 4: What is 70 - 26?
Always start with the ones place! And remember: when you regroup in subtraction, you're not making the number smaller—you're just trading 1 ten for 10 ones so you have enough to subtract.
Every digit in a number has a value based on where it sits. **Place Value Chart:** Hundreds | Tens | Ones 3 | 4 | 7 The number 347 means: • 3 hundreds = 300 • 4 tens = 40 • 7 ones = 7 **Expanded Form:** 347 = 300 + 40 + 7 **Comparing Numbers:** Start from the left (hundreds place). The number with more hundreds is greater. If hundreds are equal, compare tens. If tens are equal, compare ones.
Example 1: Write 582 in expanded form
Step 1: Identify the place value of each digit.
Step 2: The 5 is in the hundreds place: 5 × 100 = 500
Step 3: The 8 is in the tens place: 8 × 10 = 80
Step 4: The 2 is in the ones place: 2 × 1 = 2
Step 5: Expanded form: 500 + 80 + 2
Example 2: Compare 456 and 463. Which is greater?
Step 1: Line up the numbers and compare place by place.
Step 2: Compare hundreds: Both have 4 hundreds. They're equal so far.
Step 3: Compare tens: Both have 6 tens. Still equal.
Step 4: Compare ones: 6 ones vs. 3 ones. 6 > 3
Step 5: Therefore, 463 > 456
Example 3: What number is 700 + 30 + 9?
Step 1: Identify each part: 700 is hundreds, 30 is tens, 9 is ones.
Step 2: 700 means 7 in the hundreds place.
Step 3: 30 means 3 in the tens place.
Step 4: 9 means 9 in the ones place.
Step 5: The number is 739
Question 1: What is the value of the digit 6 in the number 624?
Question 2: Which number is greater: 789 or 798?
Question 3: What is 400 + 50 + 3 in standard form?
Question 4: How many tens are in 860?
Think of place value like money! 347 is like having 3 hundred-dollar bills, 4 ten-dollar bills, and 7 one-dollar bills. It helps you see why each digit's position matters.
Adding and subtracting three-digit numbers uses the same regrouping skills as two-digit numbers—we just have one more place value to work with! **Addition Strategy:** 1. Line up by place value 2. Add ones first (regroup if needed) 3. Add tens (don't forget any carried number!) 4. Add hundreds **Subtraction Strategy:** 1. Line up by place value 2. Start with ones (borrow if needed) 3. Subtract tens (borrow from hundreds if needed) 4. Subtract hundreds **Checking Your Work:** For addition, try adding in a different order. For subtraction, add your answer to the smaller number—you should get the larger number back!
Example 1: 456 + 378
Step 1: Stack the numbers, aligning place values.
Step 2: Add ones: 6 + 8 = 14. Write 4, carry 1 to tens.
Step 3: Add tens: 5 + 7 + 1 (carried) = 13. Write 3, carry 1 to hundreds.
Step 4: Add hundreds: 4 + 3 + 1 (carried) = 8
Step 5: Final answer: 834
Example 2: 625 - 347
Step 1: Stack the numbers with 625 on top.
Step 2: Ones: Can't do 5 - 7. Borrow from tens. Now 1 ten and 15 ones.
Step 3: Subtract ones: 15 - 7 = 8
Step 4: Tens: Can't do 1 - 4. Borrow from hundreds. Now 5 hundreds and 11 tens.
Step 5: Subtract tens: 11 - 4 = 7
Step 6: Subtract hundreds: 5 - 3 = 2
Step 7: Final answer: 278
Example 3: 700 - 428
Step 1: Write 700 on top and 428 below.
Step 2: We need to regroup twice! 700 = 6 hundreds, 9 tens, 10 ones
Step 3: Subtract ones: 10 - 8 = 2
Step 4: Subtract tens: 9 - 2 = 7
Step 5: Subtract hundreds: 6 - 4 = 2
Step 6: Final answer: 272
Question 1: What is 547 + 286?
Question 2: What is 812 - 456?
Question 3: What is 600 - 237?
Question 4: What is 395 + 428?
When subtracting from numbers with zeros (like 600 or 800), you might need to regroup multiple times. Take it slow and regroup one place at a time—you've got this!
Measurement helps us describe how long, tall, or wide something is. **Using a Ruler:** • Line up the zero mark with one end of the object • Read the number at the other end • Count the marks carefully! **Inches vs. Centimeters:** • Inches are bigger (1 inch ≈ 2.5 centimeters) • Centimeters are smaller and more precise **Number Lines for Measurement:** We can show measurements and solve addition/subtraction problems by jumping along a number line. Each jump represents a unit of measurement.
Example 1: A pencil measures from the 0 mark to the 5 mark on an inch ruler. How long is it?
Step 1: Find where the pencil starts: at 0
Step 2: Find where the pencil ends: at 5
Step 3: Count the spaces between: 0 to 5 is 5 spaces
Step 4: The pencil is 5 inches long
Example 2: Use a number line to solve 23 + 15
Step 1: Start at 23 on the number line
Step 2: We need to add 15, so we'll jump forward
Step 3: Jump forward 10 to land on 33
Step 4: Jump forward 5 more to land on 38
Step 5: Answer: 23 + 15 = 38
Example 3: A ribbon is 12 centimeters long. You cut off 5 centimeters. How long is it now?
Step 1: Start with 12 centimeters
Step 2: You're cutting off (subtracting) 5 centimeters
Step 3: Use a number line: start at 12, jump back 5
Step 4: 12 - 5 = 7
Step 5: The ribbon is now 7 centimeters long
Question 1: A crayon measures from 0 to 8 on a centimeter ruler. How long is the crayon?
Question 2: On a number line, you start at 45 and jump back 12. Where do you land?
Question 3: Which is longer: 10 inches or 10 centimeters?
Question 4: A string is 25 cm long. You add another piece that is 18 cm. What is the total length?
When measuring, always start at zero! And remember: the number line is your friend for visualizing addition and subtraction—it's like taking steps forward or backward.
**Telling Time to 5 Minutes:** The clock face is divided into 12 hours and 60 minutes. Each number represents 5 minutes when counting by the minute hand. • 12 = 0 minutes (top) • 1 = 5 minutes • 2 = 10 minutes • 3 = 15 minutes (quarter past) • 6 = 30 minutes (half past) • 9 = 45 minutes (quarter to the next hour) **Counting Money:** Start with the coins worth the most, then count up: 1. Quarters (25¢ each) 2. Dimes (10¢ each) 3. Nickels (5¢ each) 4. Pennies (1¢ each) **Dollar Bills:** $1 = 100 cents, $5 = 500 cents, $10 = 1,000 cents
Example 1: What time is shown when the hour hand is between 3 and 4, and the minute hand points to 7?
Step 1: The hour hand is between 3 and 4, so the hour is 3
Step 2: The minute hand points to 7
Step 3: Count by 5s: 7 × 5 = 35 minutes
Step 4: The time is 3:35
Example 2: Count these coins: 2 quarters, 1 dime, 3 nickels, 4 pennies
Step 1: Start with quarters: 2 × 25¢ = 50¢
Step 2: Add dimes: 50¢ + 10¢ = 60¢
Step 3: Add nickels: 60¢ + 5¢ + 5¢ + 5¢ = 75¢
Step 4: Add pennies: 75¢ + 1¢ + 1¢ + 1¢ + 1¢ = 79¢
Step 5: Total: 79 cents
Example 3: You have $1.00. You buy a toy for 65¢. How much change do you get?
Step 1: You start with $1.00 = 100 cents
Step 2: The toy costs 65 cents
Step 3: Subtract: 100¢ - 65¢ = 35¢
Step 4: Your change is 35 cents
Question 1: The minute hand points to 9. How many minutes is that?
Question 2: How much money is 3 dimes and 2 nickels?
Question 3: What time is it when the hour hand is between 7 and 8, and the minute hand points to 3?
Question 4: How many quarters equal one dollar?
For time, remember: the hour hand moves slowly and shows which hour we're in. The minute hand moves fast and tells us how many minutes past the hour. For money, always start counting with the biggest coins first!
**Arrays:** An array is a way to organize objects in equal rows and columns. Arrays help us see multiplication and repeated addition! Example: 3 rows of 4 stars ★★★★ ★★★★ ★★★★ This is 3 groups of 4, or 4 + 4 + 4 = 12 stars **Partitioning Shapes:** We can divide shapes into equal parts: • 2 equal parts = halves • 3 equal parts = thirds • 4 equal parts = fourths (or quarters) Each part must be the same size to be equal shares!
Example 1: Draw an array with 4 rows and 5 columns. How many objects total?
Step 1: Draw 4 rows (horizontal lines)
Step 2: Put 5 objects in each row
Step 3: Count: 5 + 5 + 5 + 5 = 20
Step 4: Or think: 4 rows × 5 in each row = 20 objects total
Example 2: A rectangle is divided into 4 equal parts. What fraction is each part?
Step 1: The whole rectangle is divided into 4 equal shares
Step 2: Each share is 1 out of 4 parts
Step 3: We write this as 1/4
Step 4: Each part is one-fourth (or one quarter) of the rectangle
Example 3: There are 3 rows of 6 apples. Write this as repeated addition and find the total.
Step 1: 3 rows means we add 6 three times
Step 2: Repeated addition: 6 + 6 + 6
Step 3: Add: 6 + 6 = 12, then 12 + 6 = 18
Step 4: Total: 18 apples
Question 1: An array has 5 rows with 3 objects in each row. How many objects total?
Question 2: A circle is divided into 2 equal parts. What is each part called?
Question 3: Which shows 4 + 4 + 4 as an array?
Question 4: A square is divided into 4 equal parts. What fraction is 1 part?
Arrays are everywhere! Think of egg cartons, muffin tins, or classroom desks. When you see equal rows and columns, you're seeing an array—and that's the beginning of understanding multiplication!