Partition Planner

Teacher Intervention Guide • HS.G-GPE.B.6

Target Group

Tier 2 Intervention (3-5 Students)

Objective

Students will find the coordinates of a point \(P\) that partitions a directed line segment \(AB\) into a given ratio \(a:b\) by applying the formula \(P = (x_1 + k(x_2 - x_1), y_1 + k(y_2 - y_1))\), where \(k = \frac{a}{a+b}\).

Common Pitfalls

Ratio vs. Fraction: Students often use the ratio \(a/b\) instead of the fraction \(a/(a+b)\).
Directionality: Students may start from the wrong endpoint (starting at \(B\) instead of \(A\)).
Negative Components: Error in calculating "Rise" or "Run" when coordinates are negative.

Required Materials

Segment Splitter Slides
Grid Guide Worksheets
Coordinate Capture Exit Ticket
Colored highlighters

Instructional Routine

1

The "Total Parts" Concept (5 min)

Use a physical or drawn string. If we want a ratio of 1:2, how many total pieces do we need? (3). This is the "Aha!" moment for the denominator.

"If I have 1 part and you have 2 parts, the whole segment must be divided into 3 equal pieces."

2

Explicit Modeling: Rise & Run (10 min)

Model finding the total horizontal distance (\(\Delta x\)) and vertical distance (\(\Delta y\)). Then, show how to multiply these by the fraction \(k\).

Point A (1, 2) to Point B (7, 11), Ratio 1:2
k = 1 / (1 + 2) = 1/3
Run = 7 - 1 = 6; Rise = 11 - 2 = 9
New Point = (1 + (1/3)(6), 2 + (1/3)(9)) = (3, 5)

3

Guided Practice with "The Grid" (15 min)

Use the Grid Guide Worksheet. Students should use highlighters: one color for the horizontal path, another for the vertical path. This visual separation reduces cognitive load.

4

Progress Monitoring (5 min)

Administer the Coordinate Capture Exit Ticket. Review immediately to identify students who are still using \(a/b\) instead of \(a/(a+b)\).

Scaffolding Cheat Sheet

Level of Support	Instructional Strategy
High Scaffold	Provide a grid with the horizontal and vertical segments already drawn as a right triangle. Use ratios that result in integer coordinates only (e.g., 1:1, 1:2 with multiples of 3).
Moderate Scaffold	Provide the formula with empty boxes for substitution. Use ratios like 2:3 or 1:4.
Low Scaffold	Give only coordinates and a ratio. Include negative coordinates or ratios that result in decimals.

Checking for Understanding Questions

"If we are partitioning a segment in a ratio of 3:2, how many equal-sized 'steps' are in the whole segment?"
"Why does the order of the points matter when the problem says 'from Point A to Point B'?"
"If our x-distance is 10 and our fraction is 2/5, how far horizontally should we move from our starting point?"

Segment Splitter Sequence

Point Partition Power

Partition Planner Teacher Guide

Segment Splitter Slides

Grid Guide Worksheet

Coordinate Capture Exit Ticket

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