Operations & Algebraic Thinking

For Parents and Teachers

This module requires adult facilitation. Pre-K students cannot read questions - you will read all prompts aloud.

Quick Guide: - Materials: 15-20 small counters (blocks, toy animals, buttons, snacks), 2 sorting mats/plates, small bowls - Time: 5-10 minutes per question - Setup: Present questions as stories, provide objects, let student physically combine/separate/count - Your role: Read aloud, demonstrate actions if needed, help count results, enter their answer - Student responds: Acts out the story with objects, counts, answers verbally - Tips: Use real situations (snacks, toys), keep numbers 1-5, make it physical, go slow

Key concepts: Addition = putting together or adding to; Subtraction = taking away or taking apart. All done with concrete objects!

Common challenges: Counting errors (model careful counting), loss of interest (shorten sessions), difficulty with "take away" (make removal very visible)

See the Math-PK course description for detailed teaching strategies.

Learning Objectives

By the end of this module, Pre-K students will be able to: - Use concrete objects to model real-world addition situations (putting together) - Use concrete objects to model real-world subtraction situations (taking away) - Solve simple addition and subtraction problems with quantities up to 5 - Understand that addition means "putting together" or "adding to" - Understand that subtraction means "taking apart" or "taking from" - Act out story problems using manipulatives - Describe what happened in addition and subtraction situations

Key Concepts

Addition as Putting Together

Young children understand addition first through physical actions: - Combining two groups of objects - Adding more objects to an existing group - Putting things together to make a larger set

At this age, addition is entirely concrete - children physically move and combine objects rather than work with abstract numbers or symbols.

Addition as Adding To

Children experience situations where a quantity increases: - "You have 2 blocks. I give you 1 more block. Now how many?" - Starting with one amount and getting more - The idea that quantities can grow

Subtraction as Taking Away

Subtraction begins with the concrete action of removal: - Starting with objects and removing some - "You had 4 crackers. You ate 2. How many are left?" - Physically seeing the before and after

Subtraction as Taking Apart

Children also experience subtraction as separation: - Breaking apart a group into two smaller groups - "You have 5 toys. Put 3 in this box and the rest in that box." - Decomposing one set into parts

Story Problems with Concrete Materials

Pre-K children solve problems by acting them out with real objects. They don't need to write equations or work abstractly - they manipulate materials to find solutions.

Prerequisites

• Basic counting skills (Module 1)
• Ability to count objects up to 5-10
• Understanding of "more" and "less"
• One-to-one correspondence when counting
• Ability to follow simple story contexts

Module Progression

Topics build from simple to more complex:

1. Putting Together (Combine) - Physically combining two small groups
2. Adding To (Result Unknown) - Starting amount + more = ?
3. Taking Away (Remove) - Starting amount - some taken = ?
4. Taking Apart (Separate) - Breaking one group into two parts
5. Story Problems with Objects - Acting out simple narratives
6. Finding How Many in Each Part - Decomposition practice
7. Zero in Addition - Adding zero means no change
8. All Taken Away - Subtracting everything leaves zero
9. Real-World Problem Solving - Applied situations
10. Explaining Solutions - Describing what you did

Real-World Connections

Addition and subtraction connect to everyday Pre-K experiences:

Addition contexts: - Snack time: "You have 2 apple slices. I give you 2 more. How many now?" - Friends joining: "3 children are playing. 1 more comes. How many now?" - Collections: "You found 2 rocks. Then you found 1 more rock." - Building: "You used 3 red blocks and 2 blue blocks. How many total?"

Subtraction contexts: - Eating snacks: "You had 5 crackers. You ate 2. How many left?" - Friends leaving: "4 children were playing. 1 went home. How many now?" - Giving away: "You have 5 stickers. You give 2 to a friend." - Breaking: "You built a tower with 4 blocks. 1 fell off."

Public-domain data connections: - NASA: "The astronauts saw 3 stars. Then they saw 2 more stars." - Nature: "There were 4 birds. 2 flew away." - Weather: "It rained 3 days. Then it rained 2 more days."

Assessment Overview

Mastery at Pre-K level looks like: - Successfully acting out story problems with objects - Counting correctly to find totals after combining or removing - Explaining actions: "I put these together" or "I took some away" - Answering "how many" questions after operations - Showing confidence with quantities 1-5 - Understanding that addition makes bigger, subtraction makes smaller (usually)

Note: Pre-K students are NOT expected to: - Write equations (2 + 3 = 5) - Memorize addition or subtraction facts - Work without concrete objects - Use abstract strategies

Teaching Strategies

Use Real, Meaningful Contexts

Every problem should connect to children's experiences: - Use familiar objects (toys, snacks, crayons) - Tell stories about things they know (friends, pets, family) - Act out situations during dramatic play

Start with Very Small Numbers

Begin with problems using 1, 2, or 3: - "You have 2 bears. You get 1 more bear." - "You had 3 crackers. You ate 1."

Gradually increase to quantities of 4 and 5.

Emphasize the Action

Focus on what's happening physically: - "Show me putting together." - "Show me taking away." - "What happened in this story?"

Concrete → Representational → Abstract (Stay Concrete!)

Pre-K students work almost entirely at the concrete level: - Concrete: Real objects you can touch and move - Representational: Pictures or drawings (minimal at this age) - Abstract: Numbers and symbols (not yet!)

Allow Multiple Solution Strategies

Children might: - Count all objects after combining - Count on from one group - Use fingers - Guess and check by counting

All strategies are valid at this stage.

Provide Many Repetitions

Children need dozens of experiences with addition and subtraction: - During centers or free play - At snack time - During transitions - In small groups - One-on-one

Mathematical Practices for Pre-K

1. Make sense of story problems: "What's happening in this story?"
2. Use tools: Manipulatives to act out problems
3. Explain thinking: "I put these with those and counted"
4. Check answers: "Let me count again to make sure"
5. Connect to experiences: "This is like when we shared the blocks"

Vocabulary

Key terms for this module: - Addition words: put together, add, plus, more, combine, total, altogether, how many in all - Subtraction words: take away, subtract, minus, less, remove, left, remaining, how many now - General: how many, count, number, equal, same - Story language: had, got, gave, lost, found, ate, used

Note: Don't emphasize formal symbols (+, -, =) at Pre-K. Focus on action words and concrete experiences.

Differentiation

For students who need support: - Use only numbers 1-3 - Provide more scaffolding ("First let's count what you have...") - Use very concrete, familiar materials - Give more time to manipulate objects - Work one-on-one or in very small groups - Accept any correct method of finding the answer

For students ready for more: - Use numbers up to 10 - Introduce problems with three addends (2 + 1 + 2) - Ask them to create their own story problems - Encourage explaining their thinking to others - Introduce simple comparison problems ("How many more?")

Parent/Caregiver Involvement

Families can support operations thinking at home: - Narrate daily activities with math language ("You had 3 carrots, you ate 1, now you have 2 left") - Play games involving combining or removing objects - Read stories that involve joining or separating - Ask "how many" questions naturally - Provide materials for acting out math stories (blocks, toys, snacks)

Connection to Mathematical Concepts

Commutativity (Informal)

Children may notice: 2 + 3 gives the same result as 3 + 2 when counting all. Don't formalize this concept, but acknowledge their observations.

Identity Property (Informal)

Adding zero doesn't change the amount. Children experience this concretely: "You have 4 blocks. No one gives you more. You still have 4."

Part-Whole Relationships

Children begin seeing that a group can be broken into parts: - 5 = 3 + 2 - 5 = 4 + 1 - 5 = 2 + 2 + 1

This foundational understanding prepares for later work with number bonds and fact families.

Sample Problem Types

Add To / Result Unknown

"Anna has 2 apples. Her mom gives her 1 more apple. How many apples does Anna have now?"

Child acts out with objects: 🍎🍎 → +🍎 → 🍎🍎🍎 (counts: 3)

Take From / Result Unknown

"Ben had 4 crackers. He ate 2 crackers. How many crackers does Ben have now?"

Child acts out: ■■■■ → removes■■ → ■■ (counts: 2)

Put Together / Total Unknown

"Cara has 2 red blocks and 3 blue blocks. How many blocks does she have altogether?"

Child combines: ■■ + ■■■ → ■■■■■ (counts: 5)

Take Apart / Addend Unknown

"David has 5 stickers. He puts some on one page and the rest on another page. How might he split them?"

Child explores multiple decompositions: - 1 and 4 - 2 and 3 - 3 and 2 - 4 and 1

Cultural Connections

Addition and subtraction situations are universal: - All cultures have stories involving combining and separating - Different cultures may use different counting objects (stones, beads, seeds) - Sharing and trading involve these operations across all societies

Safety Note

When using manipulatives: - Avoid small objects with children who still mouth items - Supervise closely during snack-based problems - Use age-appropriate, non-toxic materials

Connection to Kindergarten

This Pre-K module prepares students for Kindergarten standards: - K.OA.1: Represent addition and subtraction with objects, fingers, drawings, etc. - K.OA.2: Solve addition and subtraction word problems within 10 - K.OA.3: Decompose numbers less than or equal to 10 - K.OA.4: For any number 1-9, find the number that makes 10 - K.OA.5: Fluently add and subtract within 5

Successful Pre-K experiences with concrete operations build the foundation for more formal Kindergarten work with addition and subtraction.