Multiplication and Division, Part 1

Lesson 13

Objective

Determine the unknown whole number in a multiplication or division equation relating three whole numbers, including equations with a letter standing for the unknown quantity.

Materials and Resources

Common Core Standards

Core Standards

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  • 3.OA.A.4 — Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ÷ 3, 6 × 6 = ?.

  • 3.OA.C.7 — Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

  • 3.OA.D.8 — Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

Criteria for Success

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  1. Write all possible equations relating the same three whole numbers (e.g., for 2, 5, 10, write 2 x 5 = 10, 5 x 2 = 10, 10 $$\div$$ 2 = 5, etc.).
  2. Determine the unknown divisor in a division equation. 
  3. Determine the unknown dividend in a division equation. 
  4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers in all other cases not yet mentioned. (Spiral)
  5. Use a letter to represent an unknown quantity in an equation relating three whole numbers. 

Tips for Teachers

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  • Criteria for Success #3 is a spiral, since students have found quotients and products throughout the unit and know how to find an unknown factor since they’ve come to see its relationship to division. 
  • Solving for an unknown quantity in Grade 3 should not involve an algebraic approach, i.e., students should not be recording $$c \times 3 = 24 \rightarrow \frac{c \times 3}{3} = \frac {24}{3} \rightarrow c = 8$$. Instead, Anchor Task #1 encourages students to think of a complete fact family for three numbers that are related by multiplication and division, and then use those relationships to rewrite equations that make it easier to determine the unknown. Algebraic approaches to solving equations will come in the middle grades.
  • If you think students would benefit from some straightforward computational practice before the lesson or before jumping into the Problem Set, you can play “Multiplication Madness,” from Building Conceptual Understanding and Fluency Through Games by the North Carolina Department of Public Instruction. This game is also often referred to as “Four in a Row.” You’ll need to adapt the game so that the factors listed are 2, 3, 4, 5, and 10, and then change the sequence of products listed on the game board to be 4, 6, 8, 9, 10, 12, 15, 16, 20, 25, 30, 40, 50, and 100.
  • Students have had far more practice with finding unknowns in multiplication sentences, since they’ve come to understand division as an unknown-factor problem, but in case you’d like students to have more practice with problems of that type, they can play an analogous game to the one above called, “Missing Numbers with Multiplication,” from 3.OA.4 - About the Math, Learning Targets, and Rigor by the Howard County Public School System. Similar to above, you’ll need to eliminate facts and cards related to 0 and 1. 

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Anchor Tasks

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Problem 1

Write all of the multiplication and division equations you can think of that represent the following array. 

Guiding Questions

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Problem 2

Tehya and Kenneth are trying to figure out which number could be placed in the box to make this equation true.

Tehya insists that 50 is the only number that will make this equation true.

Kenneth insists that 2 is the only number that will make this equation true.

$$10 \div \space ? = 5$$

Who is right? Why?

Guiding Questions

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References

Illustrative Mathematics Finding the Unknown in a Division Equation

Finding the Unknown in a Division Equation, accessed on Oct. 10, 2018, 3:50 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by The Match Foundation, Inc.

Problem 3

Now, Tehya and Kenneth are trying to figure out which number could be placed in the box to make this next equation true.

Tehya insists that 20 is the only number that will make this equation true.

Kenneth insists that 5 is the only number that will make this equation true.

$$2 = \square \div 10$$

Who is right? Why? 

Guiding Questions

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References

Illustrative Mathematics Finding the Unknown in a Division Equation

Finding the Unknown in a Division Equation, accessed on Oct. 10, 2018, 3:50 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by The Match Foundation, Inc.

Problem 4

Instead of a question mark or a box, we can use a letter to represent the unknown quantity. For example, the equation in Anchor Task #3 could have been written as $$ 2 = a \div 10$$, with $$a$$ standing in for the unknown. 

Knowing that, determine the values of the unknowns that make each equation below true. 

a.  $$c\times 3=24$$

b.  $$5=50 \div g$$

c.  $$w \div 3=6$$

d.  $$24=4\times r$$

e.  $$15 \div g=3$$

f.  $$8= w \div 2$$

Guiding Questions

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Discussion of Problem Set

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  • How did you find the missing value in #4? 
  • What was the solution to the riddle?
  • How should we record our solutions when we determine the value of an unknown that’s represented with a letter? 
  • How does the relationship between multiplication and division help to determine the value of the unknowns in equations like these? 

Target Task

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Find the value of each unknown below.

1.     $$z = 5 \times 9$$

        $$z$$ = _____

2.     $$20\div v = 5$$

        $$v$$ = _____

3.     $$3 \times w = 24$$

        $$w$$ = _____

4.     $$7 = y\div 4$$

        $$y$$ = _____

References

EngageNY Mathematics Grade 3 Mathematics > Module 3 > Topic A > Lesson 3Exit Ticket, Questions #1-4

Grade 3 Mathematics > Module 3 > Topic A > Lesson 3 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by The Match Foundation, Inc.

Mastery Response

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