Use strategies and algorithms, including the standard algorithm, to multiply up to a fourdigit number by a one-digit number and to multiply a two-digit number by a two-digit

Standards Alignment

• TEKS.MATH.4.04D primary
  use strategies and algorithms, including the standard algorithm, to multiply up to a fourdigit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties;
• TEKS.MATH.4.04E primary
  represent the quotient of up to a four-digit whole number divided by a one-digit whole number using arrays, area models, or equations;
• TEKS.MATH.4.04F primary
  use strategies and algorithms, including the standard algorithm, to divide up to a fourdigit dividend by a one-digit divisor;

Lesson Overview

This Grade 4 mathematics lesson focuses on developing students' ability to multiply up to a four-digit number by a one-digit number and multiply two-digit numbers by two-digit numbers using various strategies and algorithms, including the standard algorithm. It also introduces representing division of up to a four-digit number by a one-digit number using arrays, area models, or equations and using strategies and algorithms to divide up to a four-digit dividend by a one-digit divisor. The lesson is grounded in TEKS standards and emphasizes understanding and applying mathematical properties and methods to solve problems efficiently and accurately.

Learning Objectives

• Multiply up to a four-digit number by a one-digit number using strategies such as mental math, partial products, and the standard algorithm.
• Multiply two-digit numbers by two-digit numbers using strategies including the commutative, associative, and distributive properties and the standard algorithm.
• Represent the quotient of up to a four-digit whole number divided by a one-digit whole number using arrays, area models, or equations.
• Use strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor.

Success Criteria

• Students can multiply a four-digit number by a one-digit number using at least two different strategies and explain their reasoning.
• Students can multiply two-digit numbers by two-digit numbers using the standard algorithm and other strategies.
• Students can represent division problems involving up to four-digit dividends and one-digit divisors using arrays, area models, or equations.
• Students can divide up to a four-digit number by a one-digit number using strategies and the standard algorithm with accuracy.

Prerequisite Knowledge

Students should understand place value up to the thousands place, be familiar with multiplication and division facts, and have experience with basic multiplication and division strategies such as repeated addition, arrays, and partial products.

Key Vocabulary

• Standard algorithm
• Partial products
• Commutative property
• Associative property
• Distributive property
• Array
• Area model
• Dividend
• Divisor
• Quotient

Materials and Resources

• Base-ten blocks or place value charts
• Graph paper or grid paper
• Whiteboard and markers
• Multiplication and division problem cards
• Paper and pencils

Teacher Preparation

• Prepare example problems for multiplication of four-digit by one-digit numbers and two-digit by two-digit numbers.
• Prepare visual aids for arrays and area models to represent division.
• Set up manipulatives such as base-ten blocks for hands-on activities.
• Prepare guided practice problems and exit tickets for assessment.

Detailed Lesson Notes

Understanding Multiplication Strategies

Multiplication can be performed using various strategies to build understanding and efficiency. Mental math strategies involve breaking numbers into parts that are easier to multiply, such as decomposing a number into tens and ones. Partial products involve multiplying each part separately and then adding the results. The standard algorithm is a step-by-step procedure that aligns digits by place value and multiplies systematically. The commutative property states that the order of factors does not change the product (e.g., 3 × 12 = 12 × 3). The associative property allows grouping factors differently without changing the product, and the distributive property allows breaking apart one factor to multiply separately and then add the products.

Multiplying a Four-Digit Number by a One-Digit Number

When multiplying a four-digit number by a one-digit number, students can use partial products by multiplying the one-digit number by each place value separately (thousands, hundreds, tens, ones) and then adding the results. For example, to multiply 3,245 by 6, multiply 6 × 3,000, 6 × 200, 6 × 40, and 6 × 5, then add all partial products. The standard algorithm aligns the digits and multiplies from right to left, carrying over values as needed. Modeling with base-ten blocks or place value charts helps students visualize the process and understand the value of each digit.

Multiplying Two-Digit Numbers by Two-Digit Numbers

Multiplying two-digit numbers by two-digit numbers can be done using strategies such as the area model, which breaks each number into tens and ones and multiplies each part separately before adding all products. For example, multiplying 23 by 45 involves multiplying 20 × 40, 20 × 5, 3 × 40, and 3 × 5, then adding these partial products. The standard algorithm can also be used, multiplying the ones digit of the second number by the first number, then the tens digit, and adding the results with proper place value alignment. Understanding the properties of multiplication supports these strategies.

Representing Division Using Arrays and Area Models

Division of up to a four-digit number by a one-digit number can be represented using arrays and area models to visualize how the dividend is partitioned into equal groups. An array arranges objects in rows and columns to show division as equal groups. An area model divides a rectangle into sections representing partial quotients. Equations can also represent the division process, showing the relationship between dividend, divisor, and quotient. These visual and symbolic representations help students understand division conceptually before applying algorithms.

Using Strategies and the Standard Algorithm for Division

Students use strategies such as repeated subtraction, partial quotients, and the standard long division algorithm to divide up to a four-digit dividend by a one-digit divisor. The standard algorithm involves dividing the highest place value first, subtracting the product, bringing down the next digit, and repeating until all digits have been divided. Modeling with manipulatives or drawings supports understanding of each step. Emphasizing estimation and checking work helps ensure accuracy.

Worked Examples

Classroom Activity

Use the blocks to represent each place value and multiply by the one-digit number, then add the partial products to find the total.

Guided Practice

Independent Practice

1. Foundational: Multiply 1,234 by 5 using any strategy you know.
2. Developing: Multiply 47 by 23 using the standard algorithm.
3. Application: Use an area model to multiply 56 by 34 and find the product.
4. Analysis: Explain why the distributive property helps when multiplying two-digit numbers.
5. Challenge: Divide 2,364 by 9 using the standard algorithm and explain each step.

Independent Practice Teacher Answer Key

1. 1. Using partial products: 5 × 1,000 = 5,000; 5 × 200 = 1,000; 5 × 30 = 150; 5 × 4 = 20. Sum: 5,000 + 1,000 + 150 + 20 = 6,170.
2. 2. Multiply 3 × 47 = 141; multiply 20 × 47 = 940; add 141 + 940 = 1,081.
3. 3. Break 56 into 50 + 6 and 34 into 30 + 4. Multiply: 50 × 30 = 1,500; 50 × 4 = 200; 6 × 30 = 180; 6 × 4 = 24. Add all: 1,500 + 200 + 180 + 24 = 1,904.
4. 4. The distributive property allows breaking one factor into parts (like tens and ones), multiplying each part separately, and then adding the products. This simplifies complex multiplication by turning it into easier steps.
5. 5. Divide 2,364 by 9: 9 goes into 23 two times (2 × 9 = 18), subtract 18 from 23 leaves 5; bring down 6 to make 56. 9 goes into 56 six times (6 × 9 = 54), subtract 54 leaves 2; bring down 4 to make 24. 9 goes into 24 two times (2 × 9 = 18), subtract 18 leaves 6. The quotient is 262 with a remainder of 6.

Guiding Questions

• What strategies can help us multiply large numbers?
• How do the properties of multiplication help simplify problems?
• How can arrays and area models help us understand division?
• Why is it important to understand different methods for multiplication and division?

Common Misconceptions

• Students may think the order of multiplication always changes the product, not understanding the commutative property.
• Students might add digits instead of multiplying when using the standard algorithm.
• Students may confuse place value when multiplying multi-digit numbers, leading to incorrect partial products.
• Students might think division always results in a smaller number and not understand remainders or partial quotients.
• Students may rely only on one strategy and not understand the benefits of multiple approaches.

Differentiation

Assessment

• Observe students during guided practice to check understanding of multiplication strategies.
• Ask students to explain their reasoning aloud when solving problems.
• Use exit tickets with a multiplication and division problem to assess individual mastery.
• Multiply 2,135 by 4 using any strategy and show your work.
• Divide 1,296 by 6 using an array or area model and write the quotient.
• Complete a quiz with problems multiplying four-digit by one-digit numbers and two-digit by two-digit numbers using the standard algorithm.
• Solve division problems involving up to four-digit dividends and one-digit divisors using strategies and algorithms.

Answer Guide

• Multiply 3,426 by 7 using partial products.
  Answer: 23,982
• Multiply 34 by 26 using the area model.
  Answer: 884
• Divide 1,248 by 6 using an area model or array.
  Answer: 208
• Multiply 1,234 by 5 using any strategy you know.
  Answer: 6,170
• Multiply 47 by 23 using the standard algorithm.
  Answer: 1,081

Real-Life Application

Encourage students to find examples at home where multiplication or division of large numbers is used, such as calculating total items in packages or dividing snacks among family members, and explain how they would solve the problem using strategies learned in class.

Homework or Home Connection

• Encourage students to find examples at home where multiplication or division of large numbers is used, such as calculating total items in packages or dividing snacks among family members, and explain how they would solve the problem using strategies learned in class.

Lesson Summary

In this lesson, students learn to multiply large numbers using strategies like partial products, mental math, and the standard algorithm, and multiply two-digit numbers by two-digit numbers using properties of multiplication. They also represent division problems with arrays and area models and use strategies and the standard algorithm to divide up to four-digit numbers by one-digit divisors. Through hands-on activities, visual models, and guided practice, students develop both conceptual understanding and procedural fluency in multiplication and division.

Teacher Notes

Use the exact standards alignment and retrieved-source provenance stored with this enrichment.

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