Unit 2: Operations with Whole Numbers

Learning Goals

• Use addition, subtraction, multiplication, and division with whole numbers accurately.
• Explain how each operation changes quantity and when each operation is appropriate.
• Solve one-step and multi-step word problems with clear equations and units.
• Estimate answers before calculating and check whether exact answers are reasonable.
• Connect operations through inverse relationships, such as subtraction as the inverse of addition.

Motivation / Intro

Operations are the tools we use to answer everyday questions about quantity. If we combine groups, we add. If we compare groups or remove items, we subtract. If we have equal groups, we multiply. If we split a total equally or count how many equal groups fit, we divide.

Think about a school event:

• You add snack boxes from two classrooms.
• You subtract boxes already used.
• You multiply if each table has the same number of students.
• You divide to share supplies equally.

Before doing any detailed calculation, we can estimate. For example, if we compute \(398 + 205\), we can estimate \(400 + 200 = 600\). If an exact answer is far from 600, we should re-check our work.

Intuitive Explanation

Each operation answers a different kind of question.

1. Addition asks: "How many in all?"
2. Subtraction asks: "How many left?" or "How many more?"
3. Multiplication asks: "How many total in equal groups?"
4. Division asks: "How many in each group?" or "How many groups?"

We can also view operations as pairs:

• Addition and subtraction undo each other.
• Multiplication and division undo each other.

If \(27 + 15 = 42\), then \(42 - 15 = 27\).

If \(8 \times 6 = 48\), then \(48 \div 6 = 8\).

These inverse relationships help us check answers quickly.

Formal Language / Precise Rules

Let \(a\), \(b\), and \(c\) be whole numbers.

• Addition: \(a + b = c\) means combining \(a\) and \(b\) gives total \(c\).
• Subtraction: \(a - b = c\) means removing \(b\) from \(a\) leaves \(c\), where \(a \ge b\).
• Multiplication: \(a \times b = c\) means \(a\) groups of size \(b\) make total \(c\).
• Division: \(a \div b = c\) means splitting \(a\) into groups of size \(b\) gives \(c\) groups, where \(b \ne 0\).

Useful properties:

• Commutative: \(a + b = b + a\), \(a \times b = b \times a\)
• Associative: \((a + b) + c = a + (b + c)\), \((a \times b) \times c = a \times (b \times c)\)
• Distributive: \(a \times (b + c) = a \times b + a \times c\)

Order of operations reminder:

1. Parentheses
2. Multiplication and division (left to right)
3. Addition and subtraction (left to right)

Worked Examples

Example 1: Addition with Regrouping

Compute \(4{,}786 + 2{,}597\).

Estimate first:

$$

4{,}786 \approx 4{,}800,\quad 2{,}597 \approx 2{,}600,\quad 4{,}800 + 2{,}600 = 7{,}400

$$

Exact calculation:

1. Ones: \(6 + 7 = 13\), write 3, carry 1.
2. Tens: \(8 + 9 + 1 = 18\), write 8, carry 1.
3. Hundreds: \(7 + 5 + 1 = 13\), write 3, carry 1.
4. Thousands: \(4 + 2 + 1 = 7\).

So,

$$

4{,}786 + 2{,}597 = 7{,}383

$$

The exact answer \(7{,}383\) is close to the estimate \(7{,}400\), so it is reasonable.

Example 2: Subtraction with Regrouping

Compute \(9{,}002 - 3{,}748\).

1. Ones: need \(2 - 8\), regroup from tens/hundreds/thousands.
2. After regrouping across zeros, ones become 12 and thousands decrease appropriately.
3. Ones: \(12 - 8 = 4\)
4. Tens: \(9 - 4 = 5\)
5. Hundreds: \(9 - 7 = 2\)
6. Thousands: \(8 - 3 = 5\)

Therefore,

$$

9{,}002 - 3{,}748 = 5{,}254

$$

Check with inverse operation:

$$

5{,}254 + 3{,}748 = 9{,}002

$$

So the subtraction is correct.

Example 3: Multiplication Using the Distributive Property

Compute \(36 \times 24\).

Break \(24\) into \(20 + 4\):

$$

36 \times 24 = 36 \times (20 + 4) = 36 \times 20 + 36 \times 4

$$

$$

= 720 + 144 = 864

$$

So,

$$

36 \times 24 = 864

$$

Example 4: Division with Remainder in Context

A library has 157 new books. Each shelf holds 12 books. How many full shelves can be filled, and how many books remain?

Compute:

$$

157 \div 12

$$

Since \(12 \times 13 = 156\), we get quotient 13 and remainder 1.

Interpretation:

• 13 full shelves
• 1 book left over

So,

$$

157 \div 12 = 13 \text{ remainder } 1

$$

Common Mistakes

• Using the wrong operation in word problems.

Fix: Ask what the question is really asking: combine, compare, equal groups, or sharing.

• Forgetting regrouping/carrying.

Fix: Write carries and regrouped values clearly above the place-value columns.

• Treating multiplication as repeated addition without recognizing place value.

Fix: In multi-digit multiplication, track tens and hundreds carefully.

• Ignoring remainders in context.

Fix: Decide whether to keep remainder, round up, or convert based on the real situation.

Graded Exercises

1. Easy: Compute \(2{,}459 + 3{,}786\).
2. Easy: Compute \(7{,}200 - 2{,}845\).
3. Medium: Compute \(48 \times 37\) using a clear method.
4. Medium: Compute \(945 \div 9\).
5. Medium: A store packs 126 pencils into boxes of 8. How many full boxes and how many pencils remain?
6. Hard: A class buys 24 notebooks at \(\$7\) each and 15 pens at \(\$3\) each. Find the total cost.

Exercise Solutions

1. \(2{,}459 + 3{,}786 = 6{,}245\)
2. \(7{,}200 - 2{,}845 = 4{,}355\)
3. \(48 \times 37 = 1{,}776\)
4. \(945 \div 9 = 105\)
5. \(126 \div 8 = 15\) remainder \(6\): 15 full boxes, 6 pencils remain
6. \((24 \times 7) + (15 \times 3) = 168 + 45 = 213\), total \(\$213\)

Quick Checks

1. Which operation fits: "equal groups of 9"?
2. True or false: if \(64 - 19 = 45\), then \(45 + 19 = 64\).
3. Estimate \(5{,}098 + 2{,}941\) to the nearest hundred before finding the exact sum.
4. Compute \(84 \div 7\).

Quick Check Answers

1. Multiplication
2. True
3. Estimate: \(5{,}100 + 2{,}900 = 8{,}000\); exact: \(8{,}039\)
4. 12

Summary

• Choose operations based on the meaning of the situation.
• Use place value carefully when regrouping in addition and subtraction.
• Use properties like distributive property to simplify multiplication.
• Use inverse operations and estimation to verify answers.
• Interpret division remainders in context, not just as a leftover number.