Grade 3 · Mathematics
Quarter 2 · Week 9 · TEKS
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This Grade 3 mathematics lesson focuses on helping students understand how to determine a quotient by using the relationship between multiplication and division. Students will apply strategies such as using objects, pictorial models like arrays and equal groups, and properties of operations to solve one-step and two-step problems involving multiplication and division within 100. The lesson aligns with TEKS standards TEKS.MATH.3.04J and TEKS.MATH.3.04K and emphasizes developing efficient and accurate problem-solving skills through concrete and visual representations.
Students should understand basic multiplication and division facts and be familiar with multiplication as repeated addition and division as sharing or grouping.
Multiplication and division are inverse operations. This means that multiplication can be used to check division and vice versa. For example, if you know that 4 × 5 = 20, then you can use this fact to find that 20 ÷ 5 = 4 or 20 ÷ 4 = 5. This relationship helps students determine quotients by thinking about multiplication facts they already know.
Using objects like counters or drawings such as arrays and equal groups helps students visualize division problems. For example, to solve 20 ÷ 4, students can group 20 counters into 4 equal groups and count how many are in each group. Arrays show rows and columns representing multiplication facts, which can be reversed to find division quotients.
Students will practice solving problems that require one or two steps involving multiplication and division within 100. For example, a one-step problem might be: "There are 24 apples divided equally into 6 baskets. How many apples are in each basket?" A two-step problem might involve first finding a total by multiplication and then dividing that total into groups. Teaching students to identify the operation needed and to use models or facts to solve helps build problem-solving skills.
Students can use properties like the distributive property or commutative property to break down problems into easier parts. For example, if they know 5 × 6 = 30, they can use this fact to solve related division problems. Encouraging students to recall multiplication facts supports efficient division.
Some students may confuse the roles of dividend, divisor, and quotient. Clarify these terms with examples: in 20 ÷ 4 = 5, 20 is the dividend (total amount), 4 is the divisor (number of groups), and 5 is the quotient (number in each group). Another misconception is thinking division always makes numbers smaller; explain that when dividing by a fraction or less than 1 (not covered here), the quotient can be larger, but for whole number division within 100, the quotient is smaller or equal.
Find the quotient of 36 ÷ 6 using multiplication facts.
We know that 6 × 6 = 36. Since multiplication and division are inverse operations, 36 ÷ 6 must be 6 because 6 groups of 6 make 36.
36 ÷ 6 = 6 because 6 × 6 = 36.
Solve 48 ÷ 8 by drawing equal groups.
Draw 48 counters and divide them into 8 equal groups. Count how many counters are in each group. Since 8 groups of 6 counters make 48, the quotient is 6.
48 ÷ 8 = 6 because 48 counters divided into 8 equal groups have 6 counters in each group.
Introduce a simple division problem such as 20 ÷ 4. Ask students what multiplication fact they know that relates to this division.
Students share multiplication facts they know related to the division problem, such as 4 × 5 = 20.
This activity helps students recall multiplication facts and see how they connect to division, setting the foundation for understanding quotients.
Provide counters and ask students to model 20 ÷ 4 by grouping counters into 4 equal groups and counting how many counters are in each group.
Students use counters to create 4 equal groups from 20 counters and count the number in each group to find the quotient.
Using concrete materials allows students to visualize division as sharing or grouping, reinforcing the concept of quotient.
Demonstrate how arrays represent multiplication and how they can be used to find division quotients. For example, draw a 4 by 5 array to show 4 × 5 = 20 and explain how to find 20 ÷ 4 by counting columns.
Students observe the array and explain how it shows multiplication and how it can help find the quotient in division.
Arrays visually represent multiplication facts and help students understand division as finding the number of items in each group or the number of groups.
Present one-step and two-step word problems involving multiplication and division within 100. Guide students to choose appropriate strategies such as drawing equal groups or using multiplication facts to solve.
Students solve the word problems using objects, drawings, or equations and explain their reasoning.
Applying the relationship between multiplication and division to real-world problems deepens understanding and builds problem-solving skills.
Ask students to solve a division problem using the relationship between multiplication and division and explain their thinking aloud or in writing.
Students solve the problem and describe how they used multiplication to find the quotient.
This assessment checks students’ understanding of the key concept and their ability to apply it independently.
Students use counters to create 4 equal groups from 20 counters and count the number in each group to find the quotient.
What is the quotient of 15 ÷ 3?
15 ÷ 3 = 5 because 3 × 5 = 15.
Use an array to find the quotient of 24 ÷ 4. Describe how the array helps.
An array with 4 rows and 6 columns shows 4 × 6 = 24, so 24 ÷ 4 = 6. The array helps by showing 24 objects arranged in 4 equal rows, so each row has 6 objects.
Solve this two-step problem: There are 5 baskets with 8 apples each. If the apples are divided equally into 4 bags, how many apples are in each bag?
First, multiply 5 × 8 = 40 apples total. Then divide 40 ÷ 4 = 10 apples in each bag.
Explain how knowing multiplication facts helps you solve division problems like 56 ÷ 7.
Knowing that 7 × 8 = 56 helps because division is the inverse of multiplication, so 56 ÷ 7 = 8.
A rectangle has 6 rows and 9 columns of squares. If the total number of squares is divided equally into 3 groups, how many squares are in each group? Show your reasoning.
Total squares = 6 × 9 = 54. Dividing 54 by 3 gives 54 ÷ 3 = 18 squares in each group.
Use concrete objects and step-by-step guidance to model division problems. Provide visual aids like arrays and equal groups for better understanding.
Use simple language and visuals to explain multiplication and division relationships. Encourage students to use gestures and drawings to express their thinking.
Challenge students with two-step problems involving larger numbers within 100. Encourage students to explain multiple strategies for solving division problems.
Encourage students to find examples of division and multiplication in daily life, such as sharing snacks or arranging items in equal groups, and explain how multiplication helps find the answer to division problems.
In this lesson, students learned that multiplication and division are inverse operations. They used this relationship to determine quotients by recalling multiplication facts and using concrete materials and pictorial models like arrays and equal groups. Students practiced solving one-step and two-step problems within 100, applying strategies based on objects, drawings, and properties of operations. By understanding and using these strategies, students can solve division problems more efficiently and accurately.
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