Math Medic Teacher Guide
Module 1 • Target Intervention
Math Medic Teacher Guide
Rebuilding Multiplication & Division Foundations
Focus Concept
Equal Groups & Sets
CCSS.Math.Content.3.OA.A.1-2
Conceptual Gap: Students often see multiplication as memorized facts rather than concrete aggregation. If a child cannot physically partition items or identify the difference between the number of groups and the size of each group, they will struggle.
STAGE 1
The Diagnostic Interview (Check for Understanding)
Prompt: Place 12 counting chips in a messy pile before the student. Say: "Can you show me 3 equal groups of 4 using these chips?"
| Student Response | Root Misconception | Immediate Action |
|---|
| Draws 3 lines or piles of unequal sizes (e.g., 5, 4, 3). | Lacks the concept of fairness/equality in division. | Run the Fair Share Script (below). Emphasize "equal distribution". |
| Creates 4 piles of 3 instead of 3 piles of 4. | Confuses group multiplier with group size. | Run the Container Scaffold. Explicitly label cups vs. beans. |
STAGE 2
Target Treatment Script (CPA Progression)
Phase A: Concrete (Cups and Counters) Physical
Teacher: "We have 3 cups. These are our groups. Take 12 counters. Deal them out, one-by-one, into each cup until they are all gone." | Student: (Deals 1 to each cup, repeating until 4 per cup). | Teacher: "Look inside. How many in each cup?" | Student: "4." | Teacher: "Right! 3 groups of 4 make exactly 12."
Phase B: Pictorial (Loop & Dot) Visual
Teacher: "Now draw 3 large circles on your paper. These are your groups. Draw 4 dots inside each circle. Count them by adding: 4 + 4 + 4. What is our total?" | Student: "12." | Teacher: "Perfect. 3 groups of 4 equals 12."
Phase C: Abstract (The Mapping) Equation
Teacher: "Let's use math language. The symbol \(\times\) means groups of. So, for 3 groups of 4, we write \(3 \times 4\). What is the total?" | Student: "12." | Teacher: "And if we partition 12 into 3 groups? That is division: \(12 \div 3 = 4\). The answer is our group size!"
Pro Tip: Avoid saying "times" initially. Replace with "groups of" to construct spatial meaning.
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Module 2 • Target Intervention
Math Medic Teacher Guide
Rebuilding Multiplication & Division Foundations
Focus Concept
Arrays & Area Models
CCSS.Math.Content.3.OA.A.3
Conceptual Gap: Students struggle to count systematically. An array transforms loose clusters into structured rows and columns. This spatial grouping is the physical engine that makes skip-counting and area models possible.
STAGE 1
The Diagnostic Interview (Check for Understanding)
Prompt: Provide 15 grid tiles. Say: "Can you organize these tiles to show 3 rows of 5?"
| Student Response | Root Misconception | Immediate Action |
|---|
| Builds a single long row of 15 tiles, or a random cluster. | Lacks spatial row/column structural awareness. | Use the Laser Pointer Scaffold. Track rows with an index finger. |
| Builds 3 rows and 5 columns but with misalignment/gaps. | Does not realize arrays require neat alignment to maintain structure. | Run the Grid Overlay Rescue using actual 1-inch graph paper. |
STAGE 2
Target Treatment Script (CPA Progression)
Phase A: Concrete (The Soldiery Lineup) Physical
Teacher: "Lay down 5 tiles in a straight horizontal line. That's Row 1. Now, lay down another 5 directly below them. That's Row 2. Do a 3rd row of 5 directly beneath. Tap each row: Row 1 has 5, Row 2 has 5, Row 3 has 5." | Student: (Aligns tiles carefully). | Teacher: "How many total? Skip count by fives: 5, 10, 15."
Phase B: Pictorial (Grid Shading) Visual
Teacher: "Let's color in 3 rows of 5 squares on this grid paper. Shade the first row of 5 squares, then the second row, then the third row. Outline the whole block. Let's count by columns going down: 3, 6, 9, 12, 15. The same total!"
Phase C: Abstract (The Array Law) Equation
Teacher: "In arrays, the law is: Rows \(\times\) Columns = Total. How many rows do we have?" | Student: "3." | Teacher: "And how many columns?" | Student: "5." | Teacher: "Our equation is \(3 \times 5 = 15\). Excellent job!"
Pro Tip: Physically slide an index card down an array to uncover one row at a time. This isolates the repeated addition step.
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Module 3 • Target Intervention
Math Medic Teacher Guide
Rebuilding Multiplication & Division Foundations
Focus Concept
Fact Family Relationships
CCSS.Math.Content.3.OA.A.4
Conceptual Gap: Many children view multiplication and division as isolated subjects. This script bridges them by exposing them as opposite operations acting upon the exact same three numbers (Whole, Part, Part).
STAGE 1
The Diagnostic Interview (Check for Understanding)
Prompt: Write \( 6 \times \underline{\text{ }} = 18 \) and \( 18 \div 6 = \underline{\text{ }} \) on paper. Say: "If we know \(6 \times 3 = 18\), how does that help us find the answer to \(18 \div 6\)?"
| Student Response | Root Misconception | Immediate Action |
|---|
| Guesses randomly, or starts drawing 18 dots to partition. | Sees division as a separate, complex manual computation task. | Run the Fact Triangle Rescue. Focus on inverse tracking. |
| Answers "3" but cannot explain why the two statements are linked. | Relies on procedural mimicry without structural number sense. | Run the Undo Button Script (below) using visual number bonds. |
STAGE 2
Target Treatment Script (CPA Progression)
Phase A: Concrete & Visual (The Number Bond) Physical Bond
Teacher: "Look at this Number Bond. The top circle is the Whole: 18. The bottom two are the Parts: 6 and 3. Multiplication and division are a family. Cover the 3. If we do \(18 \div 6\), who is missing?" | Student: "3." | Teacher: "Cover the 18. If we have 6 and 3, how do we find the whole?" | Student: "Multiply them: \(6 \times 3 = 18\)."
Phase B: Narrative (The Undo Button) Analogy
Teacher: "Imagine multiplication is a builder. It builds 6 groups of 3 into a tower of 18 blocks. Division is the undo button. It takes the tower of 18 and breaks it back down into 6 equal chunks. What block size do we get back?" | Student: "We get the 3-block chunks back!"
Phase C: Abstract (The Complete Family) Equation Suite
Teacher: "Let's write down the complete family code for 18, 6, and 3. There are 4 mathematical equations we can write:"
\(6 \times 3 = 18\) | \(3 \times 6 = 18\) | \(18 \div 6 = 3\) | \(18 \div 3 = 6\)
Pro Tip: Draw a visual 'Fact Triangle' on student desks using whiteboards. Use it as a quick warm-up before every small group lesson.
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