Mathematics · Grade 5 · Advanced Operations with Decimals · Term 4

Multiplying Decimals by Whole Numbers

Students will multiply decimals by whole numbers using strategies based on place value and properties of operations.

• Ontario Curriculum Expectations
• 5.NBT.B.7

About This Topic

Multiplying decimals by whole numbers extends students' whole number multiplication skills through place value understanding. Students predict the decimal point's position in products by aligning place values, such as treating 4.2 times 3 as 42 times 3 equals 126, then placing the decimal one spot from the right for 12.6. They explain similarities to whole number methods and create real-world problems, like calculating total fencing for a garden plot measured in decimal metres.

This topic supports Ontario Grade 5 expectations in number and operations, fostering fluency with decimal computations. It connects to measurement for areas and perimeters, and data management for scaling graphs. Students develop properties of operations, like the distributive property, to break down problems: 2.5 times 4 becomes (2 times 4) plus (0.5 times 4).

Active learning benefits this topic because visual models and manipulatives make place value shifts concrete. When students represent decimals with base-ten blocks or area models on grid paper, they see why the decimal point moves predictably. Peer discussions during collaborative tasks reveal thinking errors early, building confidence and precision in calculations.

Key Questions

1. Predict the placement of the decimal point in the product of a decimal and a whole number.
2. Explain how multiplying decimals is similar to multiplying whole numbers.
3. Design a real-world problem that requires multiplying a decimal by a whole number.

Learning Objectives

• Calculate the product of a decimal and a whole number using strategies based on place value.
• Explain the relationship between multiplying decimals and multiplying whole numbers, referencing properties of operations.
• Predict the correct placement of the decimal point in a product involving a decimal and a whole number.
• Design a word problem that requires multiplying a decimal by a whole number to solve.
• Compare the results of multiplying a decimal by a whole number using different strategies, such as the distributive property or standard algorithm.

Before You Start

Multiplying Whole Numbers

Why: Students need a solid understanding of the standard algorithm and properties of multiplication for whole numbers before extending these concepts to decimals.

Understanding Place Value with Decimals

Why: Students must be able to identify and understand the value of digits in decimal places (tenths, hundredths) to correctly place the decimal point in the product.

Key Vocabulary

Decimal	A number expressed using a decimal point, representing a part of a whole number based on powers of ten.
Whole Number	A non-negative integer (0, 1, 2, 3, ...) that represents a complete quantity.
Product	The result obtained when two or more numbers are multiplied together.
Place Value	The value of a digit based on its position within a number, such as ones, tens, tenths, or hundredths.
Distributive Property	A property of multiplication that states multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. For example, a(b + c) = ab + ac.

Watch Out for These Misconceptions

Common Misconception

The number of decimal places in the product equals the whole number multiplier.

What to Teach Instead

Students often miscount places based on the multiplier rather than the decimal factor alone. Using area models on grid paper shows the true product size, helping them see one decimal place stays from the original decimal. Peer teaching reinforces correct placement through shared examples.

Common Misconception

Multiplying a decimal by a whole number always makes the product smaller than the decimal.

What to Teach Instead

This stems from confusing multiplication with division. Hands-on repeated addition with manipulatives demonstrates growth for multipliers over 1, like 0.5 times 4 equals 2. Group discussions compare models to challenge this idea and build accurate expectations.

Common Misconception

Ignore place value and multiply as whole numbers without adjusting the decimal.

What to Teach Instead

Expanded form activities break decimals into wholes and parts, showing why adjustment is needed. Collaborative verification with calculators or peers catches errors, as students justify steps aloud.

Active Learning Ideas

Manipulative Match: Base-Ten Blocks

Provide base-ten flats, rods, and units to represent decimals like 0.3 or 2.4. Students multiply by a whole number using repeated addition on mats, then record the product and decimal placement. Partners verify each other's work and adjust models as needed.

35 min·Pairs

Grid Paper Arrays: Visual Multiplication

Students draw decimal rectangles on grid paper, such as 1.5 by 4 units, shading to show the decimal. They count total shaded squares to find products and predict decimal positions first. Groups share arrays on chart paper for class comparison.

40 min·Small Groups

Problem Design Relay: Real-World Scenarios

Teams line up and solve a decimal multiplication problem at the board, like 2.75 times 6 for recipe scaling. Correct answer passes baton with a new problem they create. Whole class discusses strategies after each relay.

30 min·Whole Class

Budget Builder: Shopping Simulation

Give catalogs with decimal prices. Pairs select items totaling under a budget, multiplying quantities by unit prices. They explain decimal placements and adjust selections collaboratively.

45 min·Pairs

Real-World Connections

• Calculating the total cost of multiple items when each item has a decimal price, such as a baker buying 5 kilograms of flour at $2.75 per kilogram.
• Determining the total distance traveled when a vehicle travels at a consistent decimal speed for a whole number of hours, like a train moving at 75.5 kilometers per hour for 3 hours.
• Estimating the amount of paint needed for a project where a wall's dimensions are given in decimal meters and a can covers a specific decimal area.

Assessment Ideas

Exit Ticket

Provide students with the problem: 'A recipe calls for 2.5 cups of flour per batch. If you make 4 batches, how many cups of flour do you need?' Ask students to show their work and write one sentence explaining how they determined the decimal point's placement in their answer.

Quick Check

Present students with three multiplication problems: 3.4 x 5, 0.7 x 6, and 12.1 x 2. Ask them to solve each problem and then circle the one where they felt most confident predicting the decimal point's location, writing one reason why.

Discussion Prompt

Pose the question: 'Imagine you need to multiply 1.8 by 7. How is this calculation similar to multiplying 18 by 7? How is it different? Explain your reasoning using place value.' Facilitate a brief class discussion where students share their comparisons.

Frequently Asked Questions

How do I teach multiplying decimals by whole numbers in Grade 5 Ontario math?

Start with place value charts to align digits, relating to whole number multiplication. Use strategies like expanded form: 3.2 times 5 as (3 times 5) plus (0.2 times 5). Incorporate models like decimal grids for visualization. Practice progresses from predictable problems to student-designed real-world ones, ensuring mastery of decimal placement.

What are common misconceptions when multiplying decimals by whole numbers?

Students may wrongly place the decimal based on the whole number's size or assume products are always smaller. They forget to account for the decimal factor's places only. Address with visual aids like base-ten blocks, where physical representation clarifies shifts. Regular peer explanations during activities correct these in context.

What real-world examples work for decimal by whole number multiplication?

Use cooking to scale recipes, like 1.5 cups times 4 servings; sports for 2.3 km times 5 laps; or shopping for 4.99 dollars times 3 items. Gardening plots with 2.5 m sides times perimeter multipliers connect to measurement. These contexts make place value relevant and spark problem design.

How can active learning help students master multiplying decimals by whole numbers?

Active approaches like grid paper arrays and base-ten manipulatives visualize place value shifts, making abstract rules tangible. Collaborative relays and partner budgets encourage articulating strategies, uncovering errors through discussion. These methods boost retention over worksheets, as students physically build products and defend decimal placements to peers, aligning with Ontario's emphasis on inquiry-based numeracy.