Equal Groups Planning Guide
Illustrative Mathematics Lesson Planning Protocol
Grade 3 • Unit 1 • Lesson 15
Equal Groups & Expressions
Representing multiplication situations with diagrams and equations of the form \(a \times b\)
Estimated Time: 60 Minutes
Unit & Lesson
Grade 3, Unit 1, Lesson 15
Section C: Represent & Solve Multiplication
Focus Standards
3.OA.A.1 3.OA.A.3 MP2 MP7
Teacher Learning Goal
Connect real-world situations with equal groups to visual diagrams and multiplication expressions, understanding \(a\) represents the number of groups and \(b\) the group size.
Student "I Can" Statement
"I can write multiplication expressions to match equal group situations and explain what each number means."
Required Materials
- Connecting cubes (24/pair)
- Situation & Diagram cards
- Anchor chart paper & pens
- Student math workbooks
Teacher Observations (What to Watch & Listen For)
Notice Grouping vs. Size: Do students distinguish between "how many groups" (\(a\)) and "how many in each group" (\(b\)), or do they invert the factors when drawing diagrams?
Listen for Precise Language: Listen for students stating "\(4\) groups of \(5\)" rather than simply "\(4\) and \(5\)" or immediately saying "add them up."
Warm-Up
How Many Do You See: Arrays & Equal Dots
10 mins • Routine: How Many Do You See?
Purpose
The purpose of this warm-up is to invite students to look for composite units (equal groups) within dot arrangements without counting by ones. Students use structure (MP7) to see dots grouped into rows or clusters.
Consider Asking
- • "How did you see the dots without counting each one individually?"
- • "Who grouped the dots differently? Did both groups give the same total?"
- • "How could we describe this pattern using the phrase '____ groups of ____'?"
Activity 1
Apples in Bags: Situations to Diagrams
15 mins • Whole Class Launch & Pairs
Purpose
Students interpret a concrete story situation (e.g., 4 bags with 6 apples in each bag), draw equal-group diagrams, and connect the physical story to mathematical representations. This establishes the structural convention where the first factor denotes group count.
Differentiation
Support: Provide physical counters and paper plates/cups so students can physically place 6 counters into 4 plates before sketching on paper.
Extension: Challenge students to create two different story contexts that share the same product (24) but have different group structures (\(4 \times 6\) vs. \(3 \times 8\)).
Consider Asking
- • "What in the story problem tells you the number of groups? What tells you the size of each group?"
- • "If someone drew 6 circles with 4 dots in each, does that match this exact story? Why or why not?"
- • "How does your diagram make the equal groups easy for a partner to see?"
Illustrative Mathematics Planning Guide • Grade 3 Unit 1 Lesson 15 Page 1 of 2
Lesson 15 Planning Guide (Cont.) | Activities 2 & 3, Synthesis, Assessment & Next Day Support
Grade 3 • Unit 1
Activity 2
Card Sort: Match Situations, Diagrams & Expressions
15 mins • Partner Card Sort
Purpose
Students match situation cards, visual tape/group diagrams, and multiplication expressions. This solidifies understanding of how the mathematical expression \(a \times b\) encodes "\(a\) groups of \(b\)" and builds precision with MP2 and MP7.
Differentiation
Support: Limit initial set to 3 matching sets with distinct total quantities before introducing cards that share the same product (e.g., \(2 \times 6\) and \(3 \times 4\)).
Extension: Provide blank cards for students to write a non-matching expression and challenge another pair to draw the missing diagram.
Consider Asking
- • "How did you decide which expression matched this diagram?"
- • "Look at \(3 \times 5\) and \(5 \times 3\). How do their stories and pictures differ?"
- • "What clue in the situation helped you identify the number of groups?"
Activity 3
Write Your Own Equal Groups Story
10 mins • Independent & Partner Share
Purpose
Students generate their own story context given a target multiplication expression (e.g., \(5 \times 4\)). Creating contexts reinforces contextual meaning, validates the commutative relationship in real contexts, and prepares students for independent problem solving.
Differentiation
Support: Sentence starter: "There are ____ [containers]. Each [container] has ____ [objects]. How many in all?"
Extension: Write a multi-step scenario where two different equal group sets are combined (e.g., 3 boxes of 4 pencils and 2 boxes of 5 pens).
Consider Asking
- • "Where in real life do items come in equal packages or groups?"
- • "Can your partner draw a diagram based only on your story words?"
- • "How do we know the groups in your problem are truly equal?"
Lesson Synthesis
Connecting Situations, Diagrams & Expressions
10 mins • Whole Class Debrief
Core Synthesis Prompt & Discussion:
"Today we connected situations, pictures, and expressions. When you look at the expression \(4 \times 3\), what does the 4 tell you? What does the 3 tell you?" Highlight that the first number represents the number of equal groups and the second number represents the size of each group.
Anchor Chart Addition:
Add to Class Chart: "\(a \times b\)" = \(a\) groups of \(b\) with a side-by-side illustration of 4 boxes with 3 stars in each.
Response to Student Thinking
Cool-Down Task: "A baker packs 5 boxes of muffins with 4 muffins in each box. Draw a diagram and write a multiplication expression."
Common Misconception (Addition vs. Mult):
Student writes \(5 + 4 = 9\). What it shows: Student understands the two quantities but has not yet conceptualized multiplication as equal grouping.
Inverted Factors:
Student writes \(4 \times 5\) or draws 4 boxes of 5. What it shows: Understands equal groups but interchanged group count and size.
Next Day Support
Targeted Warm-Up Adaptation: Begin Lesson 16 with a "Which One Doesn't Belong?" routine comparing: (A) 4 bags of 5 apples, (B) \(4 \times 5\), (C) \(4 + 5\), and (D) \(5 + 5 + 5 + 5\).
Small Group Center Intervention: For students who wrote addition equations, pull a 10-minute guided table using sorting cups (groups) and two-color counters to build concrete physical models before translating to expressions.
Illustrative Mathematics Planning Guide • Grade 3 Unit 1 Lesson 15 Page 2 of 2
