Word Problem Schemas Anchor Chart
1
Additive Schema • Static Sets (No Action Over Time)
Combine Schema (Part-Part-Whole)
Part + Part = Total Total − Part = Part
Anchor Story Problem
"Maya collected 18 blue sea glass gems and 14 green sea glass gems on the beach. How many sea glass gems does Maya have in all?"
How to Identify Combine
- Static Groups: Two or more distinct categories exist simultaneously without any movement or loss.
- Category Words: Look for distinct sub-groups (e.g., boys/girls, fiction/non-fiction, blue/green).
- Core Question: "Are we uniting distinct parts into one collective whole?"
Whole Unknown 18 + 14 = [ 32 ]
Part Unknown 32 − 18 = [ 14 ]
Tape Diagram: Total Unknown Find Whole
Total Whole = ? (32 gems)
Part A (Blue) 18
Part B (Green) 14
Known: Part 1 = 18 Known: Part 2 = 14 18 + 14 = 32
Variation: Part Unknown
Total (32) and Part A (18) are known. Subtract to find Part B.
32 − 18 = 14
Total Whole = 32 gems
Part A: 18
Part B = ? (14)
Key Schema Rule: In a Combine problem, the total is always the sum of all distinct parts. Subtract when finding a missing part!
Schema 1 • Anchor Chart Reference
2
Additive Schema • Dynamic Action Over Time
Change Schema (Join or Separate)
Start ± Change = End Action over Time
Anchor Story Problem (Separate Action)
"Leo had 24 toy cars in his bin. He gave some cars to his younger brother. Now Leo has 15 toy cars left. How many cars did Leo give away?"
How to Identify Change
- Action Verbs: Something happened over time (gave away, found, baked, broke, spent, received).
- Three Time Phases: 1. Start Amount → 2. Change Event → 3. Result Amount.
- Core Question: "Did an initial quantity increase (+) or decrease (−)?"
Result ? 24 − 9 = [ ? ]
Change ? 24 − [ ? ] = 15
Start ? [ ? ] − 9 = 15
Tape Diagram: Change Unknown Action: Gave Away (−)
Start Amount (Initial Total): 24 cars Start = 24
Result (Cars Kept) 15 Left
Change (Gave Away) ? (9)
Solve: 24 − 15 = 9 cars
Variation: Join Action (+)
Start with 14 cars, receive 8 more. Add to find the new result.
14 + 8 = 22
Start: 14 + Change: 8 Result: 22
Start: 14
+8 Change
Key Schema Rule: Identify whether the change increases the start (Join / Add) or decreases it (Separate / Subtract).
Schema 2 • Anchor Chart Reference
3
Additive Schema • Relational Comparison (More / Fewer)
Compare Schema (Difference / Comparison)
Larger − Smaller = Difference 2 Parallel Bars
Anchor Story Problem
"Nora read 28 books over the summer. Sam read 19 books. How many more books did Nora read than Sam?"
How to Identify Compare
- Two Separate Entities: Nora and Sam each have their own independent quantity.
- Comparison Words: "more than", "fewer than", "longer", "taller", "heavier".
- Core Question: "What is the difference or gap between the two bars?"
Find Difference: Larger − Smaller = [ ? ]
Find Larger: Smaller + Difference = [ ? ]
Find Smaller: Larger − Difference = [ ? ]
Tape Diagram: 2 Parallel Comparison Bars Difference Unknown
Nora Larger
Larger Quantity 28
Sam Smaller
Smaller: 19
Diff (?) 9 More
Equation: 28 − 19 = 9 books Nora read 9 more books than Sam.
Keyword Trap: "More" does not always mean add! If you know Sam read 19 and Nora read 9 more, you add (19 + 9 = 28). But when asking "how many more did Nora read?", you must subtract to find the difference!
Key Schema Rule: Always draw two separate bars aligned at the start to clearly see the missing difference gap.
Schema 3 • Anchor Chart Reference
4
Multiplicative Schema • Repeated Sets of Identical Size
Equal Groups Schema (Multiplication & Rate)
Groups × Group Size = Total Total ÷ Size = Groups
Anchor Story Problem
"A bakery packs 8 muffins into each box. The baker fills 5 full boxes for a morning delivery. How many muffins are packed in total?"
How to Identify Equal Groups
- Equal Containers: Every group, box, bag, pack, or team contains the exact same number of items.
- Rate Language: Key trigger words include "each", "every", "per", "a piece".
- Core Question: "Do we know both the number of groups and the size of each group?"
Find Total (Multiply) 5 × 8 = [ 40 ]
Find Groups (Divide) 40 ÷ 8 = [ 5 ]
Tape Diagram: Equal Partition Bar Total Unknown
Total Muffins = ? (40 muffins)
Box 1 8
Box 2 8
Box 3 8
Box 4 8
Box 5 8
5 equal groups (boxes) 8 items per group 5 × 8 = 40
Variation: Quotative (Measurement) Division
When total and group size are known, find number of groups: "40 muffins packed 8 per box. How many boxes?"
Find # Groups
40 Total Muffins ÷ 8 per Box = 5 Boxes
Total ÷ Group Size = # of Groups
Key Schema Rule: Equal parts mean multiplication or division! If parts are unequal, it's an additive Combine schema instead.
Schema 4 • Anchor Chart Reference
5
Multiplicative Schema • 2D Grids, Rows × Columns & Rectangular Area
Array & Area Schema (Rows × Columns)
Rows × Columns = Total Length × Width = Area
Anchor Story Problem
"Chairs in the school auditorium are set up in 6 equal rows with 12 chairs in each row. How many chairs are in the auditorium?"
How to Identify Array & Area
- Ordered 2D Spatial Structure: Items arranged in straight lines, grid intersections, or surface tiles.
- Spatial Vocabulary: "rows", "columns", "array", "square feet", "dimensions", "length × width".
- Core Question: "Does one dimension multiply the other dimension to create a surface total?"
Distributive Strategy (Break It Up!):
6 × 12 = (6 × 10) + (6 × 2)
6 × 12 = 60 + 12 = 72 chairs
Area Model / 2D Array Representation Total Chairs = ?
12 Chairs per Row (Columns)
6 Rows
Total Units (Area) 72 Chairs 6 rows × 12 chairs = 72
Vertical factor: 6 rows Horizontal factor: 12 columns Total = 72 chairs
Variation: Finding a Missing Dimension
When total area and one side are known: "72 chairs are placed in 6 equal rows. How many chairs in each row?"
Divide Total
72 Total Chairs ÷ 6 Rows = 12 Chairs / Row
Total Area ÷ Known Side = Unknown Side
Key Schema Rule: Arrays can be rotated without changing total (Commutative Property: 6 × 12 = 12 × 6).
Schema 5 • Anchor Chart Reference
6
Multiplicative Schema • Partitive Division / Equal Distribution
Fair Sharing Schema (Partitive Division)
Total ÷ # Groups = Group Size Equal Portions
Anchor Story Problem
"Carlos has 36 collectible art stickers to share equally among 4 friends. How many stickers will each friend receive?"
How to Identify Fair Sharing
- Total is Known First: You begin with a complete, large quantity of items.
- Fixed Number of Recipients: The number of friends, containers, or buckets is predetermined (4 friends).
- Core Question: "How many does each person/group get when partitioned fairly?"
Division Types Contrast:
Fair Sharing (Partitive) Know # groups (4 friends). Find size of each share (9).
Measurement (Quotative) Know size (9 per box). Find how many boxes (4).
Tape Diagram: Partitioned Unknown Share Find Share Size
Known Total = 36 Stickers
Friend 1 ? (9)
Friend 2 ? (9)
Friend 3 ? (9)
Friend 4 ? (9)
Known: 36 total stickers 4 equal shares 36 ÷ 4 = 9 each
Check with Inverse Operation
Verify division by multiplying the share size by the number of recipients: "Does each friend get 9?"
Multiply to Check
4 Friends × 9 Stickers Each = 36 Total Stickers ✓
Partitive division inverted into multiplication
Mastery Check: Always ask "Am I finding the size of each share, or the number of shares?"
Schema 6 • Anchor Chart Reference
