7日間コース
Master Linear Functions in 7 Days
Go from "I sort of get it" to "I can solve any problem on my own." Connect concepts, graphs, and real-world applications in 7 structured days.
対象: Junior High 2 / Overseas Japanese students
Turning Change into Equations
Understand that y = ax + b describes a constant rate of change. The slope (a) is how much y changes when x increases by 1. The y-intercept (b) is where the story begins.
一次関数とは「xが1増えるたびに、yが一定量だけ変わる」関係。
その「変化の割合」が傾き a。 xが0のときのyの値が切片 b。
タクシー料金で考えてみよう：
A taxi charges a base fare of 500 yen, plus 100 yen per kilometer. Express the total fare y for x kilometers.
解答
y = 100x + 500
The rate of change is 100 yen/km (slope a = 100). The starting cost is 500 yen (intercept b = 500). So the equation is y = 100x + 500.
For y = 2x + 3, find the value of y when x = 4.
解答
Substitute x = 4 into y = 2x + 3: y = 2(4) + 3 y = 8 + 3 y = 11
We replace x with 4 and compute step by step. Each step should show the calculation clearly.
Write the equation of a linear function with slope -1 and y-intercept 5.
解答
y = -x + 5
Slope a = -1, intercept b = 5. Plug directly into y = ax + b.
Reading, Drawing & Creating Graphs
Convert between equations and graphs in both directions. Given an equation, draw the graph. Given a graph, write the equation.
Two key skills:
1. Equation → Graph Start at the y-intercept (b), then use the slope (a) to find more points.
2. Graph → Equation Find the slope from two points, then find b from the y-axis crossing.
Slope formula: a = (y₂ - y₁) / (x₂ - x₁)
Example: Points (1, 5) and (3, 9) a = (9 - 5) / (3 - 1) = 4 / 2 = 2
Draw the graph of y = -2x + 4. Identify the y-intercept and one other point.
解答
y-intercept: (0, 4) When x = 1: y = -2(1) + 4 = 2 → point (1, 2) When x = 2: y = -2(2) + 4 = 0 → point (2, 0) Plot these points and draw a straight line through them.
Start at b = 4 on the y-axis. The slope is -2, so for every 1 step right, go 2 steps down.
Find the equation of the line passing through (0, 3) and (2, 7).
解答
Let the equation be y = ax + b. Find the slope: a = (7 - 3) / (2 - 0) = 4 / 2 = 2 Since the line passes through (0, 3), b = 3. Therefore y = 2x + 3
When one point is on the y-axis (x = 0), b is simply the y-coordinate of that point.
Find the equation of the line passing through (-1, 0) and (0, 2).
解答
Let the equation be y = ax + b. Find the slope: a = (2 - 0) / (0 - (-1)) = 2 / 1 = 2 From point (0, 2): b = 2 Therefore y = 2x + 2
The point (0, 2) tells us b = 2 directly. Then we just need the slope.
Building Equations from Conditions
Learn to derive the equation from any set of conditions: slope + point, two points, or graph readings.
Mixed Practice — 10 Problems
Cross-pattern problems covering equation→graph, graph→equation, two-point derivation, and word problems. Target: 20 minutes.
Word Problems — Real World to Equations
Master the framework: identify x and y, find the rate of change (slope), find the starting point (intercept), build y = ax + b.
Comprehensive Test
15 problems in 30 minutes. Section A: basics (30pts), Section B: graphs (30pts), Section C: word problems (40pts).
Solutions & Next Steps
Detailed solutions for all Day 6 problems. Common mistake patterns. Bridge to simultaneous equations and quadratic functions.