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MathematicsGrade 3· U.S. National — Common Core & NGSS
Aligned to:Common Core State Standards (Math)

Division as an Unknown-Factor Problem

Students connect multiplication and division by finding the missing factor in related equations and fact families.

Division as an Unknown-Factor Problem

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Review Related Operations

Multiplication and division are related operations. Multiplication joins equal groups to find a total, while division uses a known total to find the size or number of equal groups. Suppose 4 bags hold 6 marbles each. Multiplication shows the total: 4 × 6 = 24. If 24 marbles are shared equally among 4 bags, division finds the number in each bag: 24 ÷ 4 = 6. If the marbles are placed 6 in each bag, 24 ÷ 6 = 4 finds the number of bags. The same three numbers appear because each equation describes the same equal-group situation. You can use a multiplication fact you know to solve a related division fact.

Four bags each hold six marbles, with all twenty-four marbles shown as equal groups.
Four bags each hold six marbles, with all twenty-four marbles shown as equal groups.Source: Illustrated for this lesson

Model a Missing Factor

In division, an unknown factor is a number that is missing from a multiplication equation. To solve 28 ÷ 7, ask, “What number multiplied by 7 equals 28?” Write 7 × ? = 28. Model 28 counters in equal groups of 7. Four groups use all 28 counters, so the missing factor is 4. Therefore, 7 × 4 = 28 and 28 ÷ 7 = 4. You can also skip-count by 7: 7, 14, 21, 28. You counted four multiples, which confirms the answer. Always check by multiplying the divisor and quotient. If their product equals the dividend, the division equation is correct.

Twenty-eight counters are arranged into four equal groups of seven beside related equations.
Twenty-eight counters are arranged into four equal groups of seven beside related equations.Source: Illustrated for this lesson

Connect Multiplication and Division Equations

A multiplication equation and a division equation can tell the same story. Imagine 5 trays with 6 apples on each tray. The equation 5 × 6 = 30 gives the total number of apples. Now start with the total and ask how many apples belong on each of 5 trays. The equation becomes 30 ÷ 5 = 6. In unknown-factor form, write 5 × ? = 30. The quotient, 6, fills the missing-factor space. Notice each number’s job: 30 is the total, 5 is the known number of groups, and 6 is the size of each group. Moving between these equations helps you reason instead of guessing.

Five trays each contain six apples, showing a total of thirty apples and three related equations.
Five trays each contain six apples, showing a total of thirty apples and three related equations.Source: Illustrated for this lesson

Build Fact Families

A fact family is a group of multiplication and division equations made from the same three numbers. Use 6, 8, and 48. Since 6 × 8 = 48, changing the factor order gives 8 × 6 = 48. The related division equations are 48 ÷ 6 = 8 and 48 ÷ 8 = 6. To build a fact family, place the product, 48, first in each division equation. Then divide by one factor to find the other factor. Every equation can check another equation in the family. For example, if you solve 48 ÷ 8 = 6, check with 8 × 6 = 48. A fact family usually has four equations, but when both factors are the same, some equations repeat.

A fact-family triangle connects six, eight, and forty-eight with four equations around it.
A fact-family triangle connects six, eight, and forty-eight with four equations around it.Source: Illustrated for this lesson

Practice Unknown-Factor Problems

When you face an unknown-factor problem, first identify the total and the known factor. For 42 ÷ 6, think 6 × ? = 42. Recall multiples of 6 or draw equal groups. Because 6 × 7 = 42, the quotient is 7. For ? × 8 = 32, use the related equation 32 ÷ 8 = ?. Since 8 × 4 = 32, the missing factor is 4. Try another: 27 ÷ 3 = ?. Think 3 × ? = 27, so the answer is 9. Check every result by multiplying. If 3 × 9 equals 27, your answer fits the original total. Use arrays, skip-counting, or known facts when a solution does not come to mind right away.

A strategy chart shows three division or missing-factor problems matched with multiplication facts and equal-group models.
A strategy chart shows three division or missing-factor problems matched with multiplication facts and equal-group models.Source: Illustrated for this lesson

Complete an Exit Check

Complete each exit check without looking at the answers until you finish. First, solve 36 ÷ 9 = ?. Rewrite it as 9 × ? = 36. Second, find the missing factor in ? × 7 = 35. Third, use 4, 7, and 28 to write one multiplication equation and one related division equation. Now check your work. The first answer is 4 because 9 × 4 = 36. The second answer is 5 because 5 × 7 = 35. For the third item, one correct pair is 4 × 7 = 28 and 28 ÷ 7 = 4. If an answer was incorrect, draw equal groups and try again. You understand the lesson when you can explain how multiplication verifies each division answer.

A completed exit check shows three problems, their answers, and multiplication checks with equal groups.
A completed exit check shows three problems, their answers, and multiplication checks with equal groups.Source: Illustrated for this lesson