Mathematics · Year 7 · The Power of Number · Autumn Term

Operations with Decimals: Multiplication & Division

Mastering multiplication and division of decimals by whole numbers and other decimals.

• National Curriculum Attainment Targets
• KS3: Mathematics - Number

About This Topic

Operations with decimals centre on multiplication and division by whole numbers and other decimals. Year 7 students multiply by ignoring decimal points initially, computing the whole number product, then positioning the decimal by counting total places from both factors, such as 1.2 × 0.3 yielding 0.36. Division follows long division methods, with adjustments for decimals by multiplying numerator and denominator by powers of 10. They justify placements, compare cases like 2.4 ÷ 3 versus 3 ÷ 2.4, and predict shifts from multiplying by 10^n.

Aligned with KS3 Number standards, this builds fluency for proportional reasoning, estimation, and real-world applications like money or measurements. Students develop precision alongside mental checks for reasonableness, connecting to place value systems.

Active learning suits this topic well. Manipulatives such as decimal place value mats or money tokens make rules visible and interactive. Pair work on error detection or group relays encourages verbal justification and peer correction, turning procedures into understood concepts.

Key Questions

1. Justify the placement of the decimal point in a product of two decimals.
2. Compare dividing a decimal by a whole number to dividing a whole number by a decimal.
3. Predict the effect of multiplying a decimal by a power of 10.

Learning Objectives

• Calculate the product of two decimal numbers, justifying the placement of the decimal point by counting decimal places in the factors.
• Divide a decimal number by a whole number, accurately placing the decimal point in the quotient.
• Divide a whole number by a decimal number by converting the divisor to a whole number and performing the division.
• Compare the results of dividing a decimal by a whole number versus dividing a whole number by a decimal, explaining the differences in procedure and outcome.
• Predict and explain the effect of multiplying a decimal number by powers of 10 (10, 100, 1000) based on place value shifts.

Before You Start

Multiplication of Whole Numbers

Why: Students need a solid understanding of multiplying whole numbers before they can extend this skill to decimals.

Division of Whole Numbers

Why: Fluency with whole number division is essential for understanding the algorithms used for decimal division.

Place Value with Decimals

Why: Understanding the value of digits in decimal places is critical for correctly positioning the decimal point in products and quotients.

Key Vocabulary

Decimal point	A symbol used to separate the whole number part from the fractional part of a number. Its position indicates the value of the digits around it.
Product	The result of multiplying two or more numbers together. In decimal multiplication, the total number of decimal places in the factors determines the decimal places in the product.
Quotient	The result of dividing one number by another. In decimal division, the decimal point in the quotient aligns with its position in the dividend or is adjusted based on the divisor.
Dividend	The number that is being divided in a division problem. When dividing by a decimal, the dividend may be multiplied by a power of 10 to make the divisor a whole number.
Divisor	The number by which another number (the dividend) is divided. For decimal division, it is often converted to a whole number by multiplying both dividend and divisor by the same power of 10.

Watch Out for These Misconceptions

Common Misconception

The decimal point in a product matches the position in one factor only.

What to Teach Instead

Count total decimal places from both factors after multiplying wholes. For 0.23 × 0.4, it's three places: 0.092. Pair grid activities reveal overlaps visually, prompting students to recount and justify during discussions.

Common Misconception

Dividing a whole number by a decimal smaller than one gives a smaller answer.

What to Teach Instead

It gives a larger quotient; 5 ÷ 0.5 = 10. Bar models in small groups show partitioning wholes into tenths, clarifying the process. Peer teaching reinforces multiplying both by 10 first.

Common Misconception

Multiplying a number by 10 always moves the decimal left.

What to Teach Instead

It moves right one place. Power of 10 relays with sliders let students manipulate and observe shifts repeatedly, building accurate predictions through hands-on trial.

Active Learning Ideas

Pairs: Grid Paper Multiplication

Students draw decimals on grid paper, shading sections to represent values like 0.4 as four tenths. They overlay grids to multiply, count shaded unit squares for the product, and place the decimal accordingly. Partners explain their shading to confirm the rule.

30 min·Pairs

Small Groups: Decimal Shop Challenge

Set up a class shop with decimal-priced items. Groups receive a budget and shopping list, perform divisions to check affordability, and multiply costs for totals. Rotate buyer and cashier roles, then share strategies for decimal divisions.

40 min·Small Groups

Whole Class: Power of 10 Relay

Call students to board one by one. Display a decimal; first shifts for ×10, next for ×100, passing a marker. Class verifies with place value charts. Correct sequence wins points; discuss predictions.

25 min·Whole Class

Pairs: Division Error Hunt

Provide cards with decimal division problems and incorrect workings. Pairs identify errors, like misplaced decimals in 4.5 ÷ 1.5, correct them, and rewrite steps. Share one fix with class.

35 min·Pairs

Real-World Connections

• Retailers use decimal multiplication to calculate the total cost of multiple items, such as buying 3.5 meters of fabric at $12.75 per meter, requiring precise calculation for customer bills and inventory management.
• Engineers and construction workers use decimal division when calculating material quantities, for example, dividing the total length of a pipe (e.g., 15.75 meters) by the length of standard sections (e.g., 2.5 meters) to determine how many sections are needed.

Assessment Ideas

Exit Ticket

Provide students with two problems: 1) Calculate 2.7 x 0.4 and explain how you determined the decimal place in your answer. 2) Calculate 15.6 ÷ 3 and explain how you placed the decimal point in your answer.

Quick Check

Write 'Multiply 3.14 by 100' on the board. Ask students to write down the answer and hold up their whiteboards. Then, ask 'What is 12.5 divided by 0.5?' and have them show their answers.

Discussion Prompt

Present two division problems: 4.8 ÷ 2 and 4.8 ÷ 0.2. Ask students to work in pairs to solve both. Then, lead a class discussion: 'What was different about solving these two problems? How did changing the divisor from a whole number to a decimal affect your steps and the answer?'

Frequently Asked Questions

How do students justify decimal point placement in multiplication?

Ignore decimals, multiply as wholes, then count combined decimal places and position from the right. Example: 3.2 × 0.5 = 32 × 5 = 160, three places makes 1.60. Grids or expanded notation visualise this; estimation beforehand flags errors like 1.6.

What differs between dividing a decimal by a whole number and vice versa?

Decimal by whole uses standard division: 4.8 ÷ 3 = 1.6. Whole by decimal requires scaling both by 10 or 100 first: 3 ÷ 0.3 = 30 ÷ 3 = 10. Money contexts clarify; active scaling with counters shows equivalence and magnitude change.

How to teach predicting multiplication by powers of 10 with decimals?

Use place value charts: ×10 shifts decimal right one place, ×100 right two. Start with wholes, extend to decimals like 0.45 × 10 = 4.5. Class prediction games with reveals build pattern recognition; link to division as inverse.

How does active learning benefit decimal operations?

Active methods like manipulatives and group tasks make abstract rules concrete. Decimal mats let students physically shift points or shade products, reducing rote errors. Collaborative relays promote explaining reasoning aloud, correcting misconceptions via peer feedback. This boosts retention and confidence over worksheets alone.