Word Problems: Comparison and Two-Step

Lesson 2 of 2

In this lesson:

• Solve all three types of comparison problems using the side-by-side tape diagram
• Solve two-step word problems with two separate labeled equations
• Stay organized using a step-by-step approach

Learning Objectives

By the end of this lesson, you should be able to:

1. Represent any word problem with a labeled drawing and matching equation (from Lesson 1)
2. Solve comparison word problems involving "more than" and "fewer than" within 100
3. Solve two-step word problems within 100, writing a separate equation for each step with □ for the unknown

Comparing vs. Acting

In Lesson 1, something always happened:

• Children came out to play (adding to)
• Fish were sold (taking from)

Today: nothing happens. Two groups already exist.

"Our class has 28 books. The class next door has 19 books."

The question: "How do they compare?"

Action vs. Comparison

New diagram shape → new kind of thinking

Building the Comparison Diagram: Step 1

"Our class has 28 books. The class next door has 19 books."

• Draw the top bar for the larger amount: 28
• Draw the bottom bar for the smaller amount: 19 (shorter)
• Align both bars on the left

Building the Comparison Diagram: Step 2

"How many more books does our class have?"

• Add a bracket showing the "extra" part of the top bar
• The bracket = the difference — this is what we're finding: □

Equation: 28 − 19 = □ or 19 + □ = 28

Three Comparison Sub-Types

Which quantity is unknown? The diagram shows you — before you write the equation

Language Reversibility

The word "more" in a problem description ≠ always add

Rephrase from the other side — the diagram stays the same

Your Turn: Comparison Problems

Draw the comparison tape diagram, write the equation, and solve:

1. "Alex has 82 stickers. Alex has 25 more than Sam. How many stickers does Sam have?"
   → "More" appears — but what kind of unknown is this?

2. "A red ribbon is 64 cm long. A blue ribbon is 47 cm long. How much longer is the red ribbon?"

Draw the two bars first — then decide which equation to write

Adding a Second Step

So far: one story → one equation.

Two-step problems: one story → two events → two equations

"There were 52 birds. 17 flew away. Then 8 more landed."

• Event 1: 17 fly away → first equation → intermediate result
• Event 2: 8 land → second equation → final answer

The answer to step 1 becomes the starting point for step 2

Two Things Happen

Every two-step problem has two events:

1. First event → solve for the intermediate result (□₁)
2. Second event → use □₁ to find the final answer (□₂)

Key rule: write a separate equation for each event

Never skip the intermediate step — it shows your reasoning

Worked Example 1: Subtract Then Add

"There were 52 birds on a wire. 17 flew away. Then 8 more birds landed. How many are on the wire now?"

Step 1: 52 − 17 = □₁ = 35
Step 2: 35 + 8 = □₂ = 43

Worked Example 2: Add Then Subtract

"The art room had 38 brushes. The teacher found 14 more. Then students used 9 brushes. How many are in the art room now?"

Step 1: 38 + 14 = □₁ = 52 (teacher finds more)

Step 2: 52 − 9 = □₂ = 43 (students use some)

Circle the intermediate answer — carry it to Step 2

Staying Organized

Four-step strategy:

1. Underline the first event in the story
2. Box the second event
3. Label each equation: "Step 1:" and "Step 2:"
4. Circle the intermediate answer — use it in Step 2

Common mistake: writing "52 − 17 + 8 = □" hides the two-step structure

Always write two separate labeled equations

Your Turn: Step by Step (Problem 1)

"There were 64 children in the cafeteria. 28 went back to class. Then 15 more came in for lunch. How many children are in the cafeteria now?"

Step 1: ______ − ______ = □₁ = ______

Step 2: ______ + ______ = □₂ = ______

Label each step — circle the intermediate answer before moving to Step 2

Your Turn: Step by Step (Problem 2)

"Tomas had some books. He gave 14 to the library. His teacher gave him 6 new books. Now Tomas has 28 books. How many did he start with?"

Start unknown — work backwards from the end!

Step 2 first: 28 − 6 = ______ (he had this many after giving books away)

Step 1: ______ + 14 = ______ (he started with this many)

Key Takeaways

✓ Comparison problems use two side-by-side bars with a bracket for the difference
✓ Three comparison sub-types: difference unknown, larger unknown, smaller unknown
✓ Two-step problems need two separate labeled equations — one per event
✓ Circle the intermediate answer and carry it forward to the next step

Keywords mislead: draw the tape diagram before writing any equation
Word order misleads: the longer bar is always the whole — subtract the shorter from it
Two-step = two equations: never combine into one expression

You've Completed 2.OA.A.1

What you can now do:

• Identify all five situation types and set up the correct equation
• Solve with the unknown in any position — result, change, or start
• Solve comparison problems using the side-by-side tape diagram
• Solve two-step problems with two separate labeled equations

Next: 2.OA.A.2 — Fluency with addition and subtraction within 20

The word problem skills you built in both lessons will be used every year from now on!