Put Together & Take Apart Problems - Second Grade Mathematics

Unlike dynamic change problems where an action occurs over time, static situations involve collections of objects that simply exist together simultaneously. You might have a vase containing a mix of red roses and yellow daisies, a fruit bowl holding apples and bananas, or a pencil case with wooden pencils and markers. These are known as Put Together and Take Apart problems, governed by the timeless part-part-whole relationship. In this chapter, you will master the two major variations—Total Unknown and Addend Unknown—and learn how to model static scenarios using clean visual bar models.

The Part-Part-Whole Architecture

In Put Together and Take Apart problems, there is no time lapse, no moving story, and no before-and-after. Two distinct subgroups coexist to form one combined collection.

Static Part-Part-Whole Relationship:
+-------------------------------------------------------+
|                    WHOLE (TOTAL)                      |
+---------------------------+---------------------------+
|          PART A           |          PART B           |
+---------------------------+---------------------------+

Equation: PART A + PART B = TOTAL (WHOLE)

The Two Subgroups

Consider a classroom aquarium: Part A: 16 Neon Tetra fish Part B: 19 Guppy fish Whole: Total fish swimming in the aquarium Whether you think of putting the two groups together into one tank, or taking the tank collection apart into its two species, the underlying numbers remain in exact balance.

Variation 1: Total Unknown

In Total Unknown problems, you are given the sizes of both parts and asked to determine the combined total whole.

Total Unknown Example:
Story: In a flower bed, there are 34 red tulips and 28 white tulips. How many tulips are growing in the flower bed in all?

Bar Model:
+-------------------------------------------------------+
|                       TOTAL: [ ? ]                    |
+---------------------------+---------------------------+
|      34 Red Tulips        |      28 White Tulips      |
+---------------------------+---------------------------+

Equation: 34 + 28 = [ ? ]
Calculation: 34 + 20 = 54, plus 8 = 62 tulips.

Identifying Total Unknown Clues

Look for questions that ask: "How many in all?", "How many altogether?", or "What is the total number of items?" Because both parts are known, your mathematical job is to join them using addition.

Variation 2: Both Addends Unknown and One Addend Unknown

In One Addend Unknown problems, you know the grand total and one subgroup, but you need to determine the size of the other subgroup.

One Addend Unknown Example:
Story: A pet store has 45 reptiles in total. 27 of the reptiles are turtles, and the rest are lizards. How many lizards are in the store?

Bar Model:
+-------------------------------------------------------+
|                   TOTAL: 45 REPTILES                  |
+---------------------------+---------------------------+
|        27 Turtles         |       [ ? ] Lizards       |
+---------------------------+---------------------------+

Equation: 27 + [ ? ] = 45   (or 45 - 27 = [ ? ])
Calculation: 45 - 27 = 18 lizards.

Why Subtraction Solves Addend Unknown

Even though nobody took any lizards away from the store, you use subtraction to solve the problem! Subtraction separates the total whole (45) from the known subgroup (27 turtles), leaving behind the unknown subgroup (18 lizards).

Both Addends Unknown (Decomposing a Whole)

Sometimes a problem gives you a total, such as 10 flowers, and asks: "How many could be red and how many could be yellow?" This open-ended variation invites you to explore all combinations that add up to 10: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5.

Practice Exercises

Exercise 1: Determine whether this problem is Total Unknown or Addend Unknown, write a matching equation, and solve: "A fruit stand has a crate containing 65 pieces of citrus fruit. 38 of the fruits are oranges, and the rest are grapefruits. How many grapefruits are in the crate?"

Exercise 2: Solve this problem: "On a playground, 29 children are playing soccer and 34 children are playing on the swings. How many children are playing on the playground altogether?"

Exercise 3: Draw or describe a bar model that shows how the number 50 can be decomposed into two unequal parts where one part is 32. What is the missing part?

Exercise 4: Explain why a story where nothing is lost, sold, or eaten can still be solved using the subtraction operation.

Solutions and Explanations

Solution to Exercise 1: This is an Addend Unknown problem because the total is given (65 fruits) along with one part (38 oranges), and the other part (grapefruits) is unknown. Equation: 38 + ? = 65 (or 65 - 38 = ?). Calculating: 65 - 30 = 35, and 35 - 8 = 27. There are 27 grapefruits in the crate.

Solution to Exercise 2: This is a Total Unknown problem because both parts are known (29 and 34) and the combined total is requested. Equation: 29 + 34 = ?. Add tens: 20 + 30 = 50. Add ones: 9 + 4 = 13. Combine: 50 + 13 = 63 children playing altogether.

Solution to Exercise 3: Draw a top rectangular bar labeled Total: 50. Below it, draw two adjacent boxes: the left box is labeled 32, and the right box is labeled with a question mark. To find the missing part, calculate 50 - 32 = 18. The missing part is 18.

Solution to Exercise 4: Subtraction has two distinct mathematical meanings: taking away and separating a whole into parts. Even when no items are physically removed or destroyed over time, subtraction is the mathematical tool used to separate a known subgroup from a grand total to find the size of the remaining subgroup.