Mathematics · Grade 4 · Multiplicative Thinking and Operations · Term 1

Multiplying by One-Digit Numbers

Students multiply a whole number of up to four digits by a one-digit whole number using various strategies including the standard algorithm.

• Ontario Curriculum Expectations
• CCSS.MATH.CONTENT.4.NBT.B.5

About This Topic

Division in Grade 4 is explored through the lens of partitioning and fair sharing. Students move beyond basic facts to divide two-digit numbers by one-digit numbers, often encountering remainders for the first time. The Ontario curriculum emphasizes that a remainder isn't just a leftover number; its meaning changes based on the real-world context. For example, if 13 students are going on a trip in cars that hold 4 people, you need 4 cars, not 3.25.

Students also learn the inverse relationship between multiplication and division, using their knowledge of factors to solve problems. This topic is highly practical and benefits from simulations where students must distribute resources fairly. Students grasp this concept faster through structured discussion and peer explanation, particularly when debating how to handle a remainder in different scenarios.

Key Questions

1. Design a strategy to multiply a four-digit number by a one-digit number.
2. Evaluate the efficiency of different multiplication strategies (e.g., area model vs. standard algorithm).
3. Predict the product of a multi-digit number and a one-digit number using estimation.

Learning Objectives

• Calculate the product of a four-digit whole number and a one-digit whole number using the standard algorithm.
• Compare the efficiency of the area model and the standard algorithm for multiplying multi-digit numbers by one-digit numbers.
• Estimate the product of a four-digit number and a one-digit number to predict the approximate size of the answer.
• Explain the steps involved in the standard algorithm for multiplication, including regrouping.
• Apply multiplication strategies to solve word problems involving quantities up to four digits multiplied by a one-digit number.

Before You Start

Multiplication Facts to 10x10

Why: Students need automatic recall of basic multiplication facts to perform calculations within the standard algorithm and area model efficiently.

Place Value to Thousands

Why: Understanding the value of digits in the thousands, hundreds, tens, and ones places is crucial for correctly applying the standard algorithm and interpreting partial products.

Introduction to Area Model

Why: Familiarity with representing multiplication visually using the area model provides a foundation for understanding larger multiplication problems and comparing strategies.

Key Vocabulary

Standard Algorithm	A step-by-step procedure for multiplying numbers, which involves multiplying digits in specific place value positions and regrouping when necessary.
Area Model	A visual representation of multiplication where the factors are represented as the dimensions of a rectangle, and the product is the area of the rectangle.
Partial Products	The products obtained from multiplying parts of the numbers being multiplied, which are then added together to find the final product.
Regrouping	The process of exchanging a quantity from one place value to another, such as exchanging 10 ones for 1 ten, when performing addition or multiplication.

Watch Out for These Misconceptions

Common Misconception

Thinking the remainder can be larger than the divisor.

What to Teach Instead

Students often stop dividing too early. Use physical counters to show that if you have 5 left over and you are dividing by 4, you can still give everyone one more piece. Peer checking helps catch this error.

Common Misconception

Believing division is always 'making things smaller' without context.

What to Teach Instead

While the quotient is smaller than the dividend, students need to see division as a way to organize. Use sharing simulations to show that division is about creating equal groups, not just reducing a number.

Active Learning Ideas

Simulation Game: The Feast Coordinator

Students are given a set number of 'bannock pieces' (counters) and must divide them among different numbers of guests. They must decide what to do with remainders in different contexts: cutting them into fractions, giving them away, or needing an extra plate.

35 min·Small Groups

Stations Rotation: Division Strategies

Set up stations for different division methods: one for repeated subtraction, one for using arrays, and one for partial quotients. Students solve the same problem at each station to see which method feels most efficient for them.

45 min·Small Groups

Think-Pair-Share: The Remainder Riddle

Give a word problem where the remainder must be handled differently (e.g., 'rounding up' for buses vs. 'ignoring' for leftover change). Students discuss their decision with a partner and justify why their answer makes sense in the real world.

20 min·Pairs

Real-World Connections

• Event planners use multiplication to calculate the total number of chairs needed for a large banquet, multiplying the number of tables by the number of seats per table, which could be a four-digit number of guests.
• Retailers estimate inventory needs by multiplying the number of items expected to sell per day by the number of days in a sales period, for example, calculating how many of a popular toy might be sold in a month.
• Construction workers might calculate the total amount of concrete needed for a large foundation by multiplying the volume per section by the number of sections, potentially involving large numbers.

Assessment Ideas

Quick Check

Present students with the problem: 'A school orders 1,234 pencils for each of its 4 grades. How many pencils are ordered in total?' Ask students to solve it using the standard algorithm and show their work. Check for correct calculation and understanding of place value.

Discussion Prompt

Pose the question: 'Imagine you need to multiply 3,456 by 7. Which strategy would you choose, the area model or the standard algorithm, and why? Explain the advantages and disadvantages of each for this specific problem.'

Exit Ticket

Give students a multiplication problem, such as 2,500 x 5. Ask them to first estimate the product, then calculate the exact product using any strategy they prefer. Collect their responses to gauge understanding of both estimation and calculation.

Frequently Asked Questions

How can active learning help students understand division?

Active learning, like the 'Feast Coordinator' simulation, puts division into a meaningful context. When students physically move objects into groups, they see the 'fairness' of division. Discussing remainders in groups helps them understand that math isn't just about numbers; it's about making decisions. This social interaction helps clarify the relationship between the dividend, divisor, and quotient more effectively than a worksheet.

What is the best way to explain a remainder?

Explain a remainder as 'what is left over' when you cannot make another equal group. Always use a context, like leftover stickers or extra seats, to help students decide if they should ignore it, round up, or turn it into a fraction.

How does division relate to multiplication in Grade 4?

They are inverse operations. In Grade 4, we teach students to use 'related facts.' If they know 4 x 6 = 24, they automatically know 24 ÷ 4 = 6. This connection is vital for mental math and checking work.

What is 'partial quotients'?

Partial quotients is a strategy where students take away 'easy' chunks of the divisor from the total. For example, to solve 75 ÷ 5, a student might take away 50 (10 groups) and then 25 (5 groups) to get 15. It's a precursor to long division that builds better number sense.