Add To & Take From Word Problems - Second Grade Mathematics
Mathematics is the language of action, and in real life, quantities are constantly changing over time: birds join a flock on a branch, leaves blow away from a pile, coins drop into a piggy bank, or pages are read from a book. These dynamic situations are known as change problems, classified into two major families: Add To problems, where an action increases the quantity, and Take From problems, where an action decreases the quantity. In this chapter, you will learn how to analyze the three phases of any change problem—the start, the change, and the result—and discover how to solve for any unknown piece no matter where it hides in the story.
The Three Phases: Start, Change, and Result
Every dynamic word problem tells a short three-act mathematical story.
The Anatomy of a Change Problem:
+----------------+ +-----------------+ +-----------------+
| START | ---> | CHANGE | ---> | RESULT |
| Initial Amount | | Added or Taken | | Final Amount |
+----------------+ +-----------------+ +-----------------+
Add To Equation: START + CHANGE = RESULT
Take From Equation: START - CHANGE = RESULT
The Three Unknown Variations
Any one of these three parts can be the mystery number you are asked to discover: Variation 1: Result Unknown (The most familiar: you know the start and change, find the result). Variation 2: Change Unknown (You know how many were there at first and how many were there at the end, find what happened in the middle). Variation 3: Start Unknown (The most challenging: you do not know how many were there at the beginning, but you know what changed and how many ended up there).
Mastering "Add To" Problem Types
In an Add To problem, an action causes the starting amount to increase over time.
Type 1: Result Unknown
Story: Leo had 28 baseball cards. His uncle gave him 15 more cards. How many cards does Leo have now?
Model: 28 (Start) + 15 (Change) = [ ? ] (Result)
Calculation: 28 + 15 = 43 cards.
Type 2: Change Unknown
Story: Leo had 28 cards. His uncle gave him some more cards. Now Leo has 43 cards. How many cards did his uncle give him?
Model: 28 + [ ? ] = 43
Calculation: Flip to subtraction! 43 - 28 = 15 cards.
Type 3: Start Unknown
Story: Leo had some cards. His uncle gave him 15 more cards. Now Leo has 43 cards. How many cards did Leo have at first?
Model: [ ? ] + 15 = 43
Calculation: Flip to subtraction! 43 - 15 = 28 cards.
Transforming Change and Start Unknowns
Notice that when the unknown sits in the middle (Change Unknown) or at the beginning (Start Unknown), you do NOT add! You use subtraction to find the difference between the result and the known part.
Mastering "Take From" Problem Types
In a Take From problem, an action causes the starting amount to decrease over time.
Type 1: Result Unknown
Story: A bakery had 54 cookies. Customers bought 18 cookies. How many cookies were left?
Model: 54 (Start) - 18 (Change) = [ ? ] (Result)
Calculation: 54 - 18 = 36 cookies.
Type 2: Change Unknown
Story: A bakery had 54 cookies. Customers bought some cookies. Then there were 36 cookies left. How many cookies were bought?
Model: 54 - [ ? ] = 36
Calculation: Subtract the remaining amount from the start! 54 - 36 = 18 cookies.
Type 3: Start Unknown
Story: A bakery had some cookies. Customers bought 18 cookies. Then there were 36 cookies left. How many cookies did the bakery have at first?
Model: [ ? ] - 18 = 36
Calculation: Add the parts back together! 36 + 18 = 54 cookies.
The Start-Unknown Trap in Take From
Pay special attention to Start Unknown in Take From problems: people hear the word "bought" or "left" and want to subtract. But think about the story: before anyone bought any cookies, there had to be MORE cookies in the shop than at the end! To find the larger starting whole, you must ADD the cookies that were bought to the cookies that were left over: 36 + 18 = 54.
Practice Exercises
Exercise 1: Identify whether this problem is Add To or Take From, state which piece is unknown (start, change, or result), and solve: "Sofia had 35 stickers. Her sister gave her some more stickers. Now Sofia has 52 stickers. How many stickers did her sister give her?"
Exercise 2: Solve this Start Unknown problem: "A squirrel had buried some acorns. A crow dug up and stole 14 acorns. Now the squirrel has 28 acorns left buried. How many acorns did the squirrel bury at first?"
Exercise 3: Write an addition or subtraction equation for this story and solve: "There were 62 passengers on a morning train. At the city station, 27 passengers got off. How many passengers remained on the train?"
Exercise 4: Explain why you must add, rather than subtract, to find the answer to a Take From problem where the start is unknown.
Solutions and Explanations
Solution to Exercise 1: The story is an Add To problem because stickers were given to Sofia, increasing her collection. The change is unknown (Sofia received "some more"). Equation: 35 + ? = 52. To find the unknown change, subtract: 52 - 35 = 17. Her sister gave her 17 stickers.
Solution to Exercise 2: This is a Take From problem with Start Unknown. Equation: ? - 14 = 28. To find how many acorns were there at the beginning before any were stolen, add the remaining acorns to the stolen acorns: 28 + 14 = 42. The squirrel had buried 42 acorns at first.
Solution to Exercise 3: This is a Take From problem with Result Unknown. Equation: 62 - 27 = ?. Subtracting: 62 - 20 = 42, and 42 - 7 = 35. Exactly 35 passengers remained on the train.
Solution to Exercise 4: In a Take From problem, the starting amount represents the entire whole collection before anything was removed. What was removed is one part, and what remains is the other part. Because the whole equals the sum of its parts, you must combine the amount taken away with the amount left over to reconstruct the larger starting amount.