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line_graph_intuition.html
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line_graph_intuition.html
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<!DOCTYPE html>
<html data-require="math math-format expressions graphie graphie-helpers interactive word-problems">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Line graph intuition</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="graph-slope" data-weight="1">
<div class="vars">
<var id="SLOPE">randFromArray([
"positive", "negative", "zero", "undefined"
])</var>
<var id="POINTS">{}</var>
</div>
<p class="question">
Graph a line that has <var>an(SLOPE)</var> slope:
</p>
<div class="graphie" id="grid">
graphInit({
range: 11,
scale: 20,
axisArrows: "<->",
tickStep: 1,
labelStep: 1,
gridOpacity: 0.05,
axisOpacity: 0.2,
tickOpacity: 0.4,
labelOpacity: 0.5
});
label([0, 11], "y", "above");
label([11, 0], "x", "right");
addMouseLayer();
POINTS.pointA = addMovablePoint({
coord: [-5, 5],
snapX: 0.5,
snapY: 0.5,
});
POINTS.pointB = addMovablePoint({
coord: [5, 5],
snapX: 0.5,
snapY: 0.5,
});
graph.line1 = addMovableLineSegment({
pointA: POINTS.pointA,
pointZ: POINTS.pointB,
fixed: true,
extendLine: true,
normalStyle: {
stroke: ORANGE,
"stroke-width": 2
}
});
// A and B can't be in the same place
POINTS.pointA.onMove = function(x, y) {
return (x !== POINTS.pointB.coord[0] ||
y !== POINTS.pointB.coord[1]);
}
POINTS.pointB.onMove = function(x, y) {
return (x !== POINTS.pointA.coord[0] ||
y !== POINTS.pointA.coord[1]);
}
POINTS.pointA.toFront();
POINTS.pointB.toFront();
</div>
<div class="solution" data-type="custom">
<div class="instruction">
Drag the two points to move the line into the correct
position.
</div>
<div class="guess">
[POINTS.pointA.coord, POINTS.pointB.coord]
</div>
<div class="validator-function">
if (SLOPE !== "zero" &&
guess[0][0] === -5 &&
guess[0][1] === 5 &&
guess[1][0] === 5 &&
guess[1][1] === 5) {
return "";
}
var slope = (guess[1][1] - guess[0][1]) /
(guess[1][0] - guess[0][0]);
if (SLOPE === "positive") {
return slope > 0;
} else if (SLOPE === "negative") {
return slope < 0;
} else if (SLOPE === "zero") {
return abs(slope) < 0.001;
} else if (SLOPE === "undefined") {
return guess[1][0] === guess[0][0];
}
</div>
<div class="show-guess">
POINTS.pointA.setCoord(guess[0]);
POINTS.pointB.setCoord(guess[1]);
graph.line1.transform(true);
</div>
</div>
<div class="hints">
<p>
The slope of a line is the ratio of the change in
<code>y</code> over the change in <code>x</code>. Also
known as rise over run.
</p>
<p data-if="SLOPE === 'positive'">
For a positive slope, if the change in <code>y</code> from
one point to another is positive then the change in
<code>x</code> should also be positive.
</p>
<p data-else-if="SLOPE === 'negative'">
For a negative slope, if the change in <code>y</code> from
one point to another is positive then the change in
<code>x</code> should be negative.
</p>
<p data-else-if="SLOPE === 'zero'">
For a zero slope, the change in <code>y</code> should be
zero from any point to another point.
</p>
<p data-else>
For an undefined slope, the change in <code>x</code> should
be zero from any point to another point.
</p>
<div>
<p>
A line with <var>an(SLOPE)</var> slope looks like:
</p>
<div class="graphie" style="margin-left: 30px">
graphInit({
range: 6,
scale: 20,
axisArrows: "<->",
tickStep: 1,
labelStep: 10,
gridOpacity: 0.05,
axisOpacity: 0.2,
tickOpacity: 0.4,
labelOpacity: 0.5
});
style({ stroke: ORANGE }, function() {
var intercept = randRange(-3, 3);
if (SLOPE === "positive") {
plot(function(x) {
return x + intercept;
}, [-6, 6]);
} else if (SLOPE === "negative") {
var yInt = randRange(-3, 3);
plot(function(x) {
return -x + intercept;
}, [-6, 6]);
} else if (SLOPE === "zero") {
var y = randRangeNonZero(-3, 3);
path([[-6, intercept], [6, intercept]]);
} else if (SLOPE === "undefined") {
var x = randRangeNonZero(-3, 3);
path([[intercept, -6], [intercept, 6]]);
}
});
</div>
</div>
</div>
</div>
<div id="graph-compare" data-type="graph-slope" data-weight="1">
<div class="vars">
<var id="SLOPE_N">randRangeNonZero(-5, 5)</var>
<var id="SLOPE_D">randRange(1, 5)</var>
<var id="Y_INT">randRange(-5, 5)</var>
<var id="GOAL">SLOPE_N < 0 ? "negative" : "positive"</var>
<var id="CHANGE">randFromArray(["greater", "less"])</var>
<var id="NEW_SLOPE">(function() {
if (SLOPE_N < 0) {
if (CHANGE === "greater") {
return (SLOPE_N / SLOPE_D) / 2;
} else {
return (SLOPE_N / SLOPE_D) * 2;
}
} else {
if (CHANGE === "less") {
return (SLOPE_N / SLOPE_D) / 2;
} else {
return (SLOPE_N / SLOPE_D) * 2;
}
}
})()</var>
</div>
<p class="question">
Graph a line that has a slope that is <em><var>GOAL</var></em>
and <em><var>CHANGE</var> than</em> the slope of the blue line.
</p>
<div class="graphie" data-update="grid">
plot(function(x) {
return (SLOPE_N / SLOPE_D) * x + Y_INT;
}, [-11, 11], {
stroke: BLUE
}).toBack();
</div>
<div class="solution" data-type="custom">
<div class="instruction">
Drag the two points to move the line into the correct
position.
</div>
<div class="guess">
[POINTS.pointA.coord, POINTS.pointB.coord]
</div>
<div class="validator-function">
if (guess[0][0] === -5 &&
guess[0][1] === 5 &&
guess[1][0] === 5 &&
guess[1][1] === 5) {
return "";
}
var slope = (guess[1][1] - guess[0][1]) /
(guess[1][0] - guess[0][0]);
if (GOAL === "positive" &&
CHANGE === "less") {
return (slope > 0) &&
(slope < SLOPE_N / SLOPE_D);
} else if (GOAL === "positive" &&
CHANGE === "greater") {
return (slope > 0) &&
(slope > SLOPE_N / SLOPE_D);
} else if (GOAL === "negative" &&
CHANGE === "less") {
return (slope < 0) &&
(slope < SLOPE_N / SLOPE_D);
} else if (GOAL === "negative" &&
CHANGE === "greater") {
return (slope < 0) &&
(slope > SLOPE_N / SLOPE_D);
}
</div>
<div class="show-guess">
POINTS.pointA.setCoord(guess[0]);
POINTS.pointB.setCoord(guess[1]);
graph.line1.transform(true);
</div>
</div>
<div class="hints">
<p>
The slope of the original line is
<span data-if="SLOPE_N > 0">positive.</span>
<span data-else>negative.</span>
</p>
<div data-if="GOAL === 'negative'">
<div data-if="CHANGE === 'greater'">
<p>
If the slope of the new line is negative and
greater than the slope of the original line with
negative slope then the new line should be less
steep.
</p>
</div>
<div data-else>
<p>
If the slope of the new line is negative and less
than the slope of the original line with negative
slope, then the new line should be steeper.
</p>
</div>
</div>
<div data-else>
<div data-if="CHANGE === 'greater'">
<p>
If the slope of the new line is positive and greater
than the slope of the original line with positive
slope, then the new line should be steeper.
</p>
</div>
<div data-else>
<p>
If the slope of the new line is positive and less
than the slope of the original line with positive
slope, then the new line should be less steep.
</p>
</div>
</div>
<div>
<p>
Since the original slope is <code>
<var>fractionReduce(SLOPE_N, SLOPE_D)</var></code>,
the slope of the new line could be <code>
<var>decimalFraction(NEW_SLOPE, true, true)</var>
</code> and could look like this:
<p>
<div class="graphie" style="margin-left: 30px">
graphInit({
range: 11,
scale: 10,
axisArrows: "<->",
tickStep: 1,
labelStep: 20,
gridOpacity: 0.05,
axisOpacity: 0.2,
tickOpacity: 0.4,
labelOpacity: 0.5
});
style({ stroke: BLUE }, function() {
plot(function(x) {
return x * (SLOPE_N / SLOPE_D) + Y_INT;
}, [-11, 11]);
});
style({ stroke: ORANGE }, function() {
plot(function(x) {
return x * NEW_SLOPE + 2;
}, [-11, 11]);
});
</div>
</div>
</div>
</div>
<div id="find-slope-and-yint" data-weight="1">
<div class="vars" data-ensure="!(M === 1 && B === 0) &&
abs(M) < 10 && abs(B) < 10 && INCR < 6">
<var id="X1, X2">randRangeUnique(-8, 8, 2)</var>
<var id="Y1, Y2">randRangeUnique(-8, 8, 2)</var>
<var id="RISE">Y2 - Y1</var>
<var id="RUN">X2 - X1</var>
<var id="M">RISE / RUN</var>
<var id="B">Y1 - X1 * M</var>
<var id="INCR">abs(RUN) / getGCD(abs(RISE), abs(RUN))</var>
<var id="GRAPH">{}</var>
</div>
<p class="question">
Adjust the slope and y-intercept until it connects the two
points. How do the slope and y-intercept values affect the line?
</p>
<div class="graphie" id="grid">
graphInit({
range: 11,
scale: 20,
axisArrows: "<->",
tickStep: 1,
labelStep: 1,
gridOpacity: 0.05,
axisOpacity: 0.2,
tickOpacity: 0.4,
labelOpacity: 0.5
});
label([0, 11], "y", "above");
label([11, 0], "x", "right");
style({
stroke: BLUE,
fill: BLUE
}, function() {
circle([X1, Y1], 0.2);
circle([X2, Y2], 0.2);
});
GRAPH.MN = 1;
GRAPH.MD = 1;
GRAPH.BN = 1;
GRAPH.BD = 1;
var plot = line([-11, -11 * GRAPH.MN / GRAPH.MD +
GRAPH.BN / GRAPH.BD], [11, 11 * GRAPH.MN / GRAPH.MD +
GRAPH.BN / GRAPH.BD], { stroke: BLUE });
GRAPH.redraw = function() {
plot.remove();
plot = line([-11, -11 * GRAPH.MN / GRAPH.MD +
GRAPH.BN / GRAPH.BD], [11, 11 * GRAPH.MN / GRAPH.MD +
GRAPH.BN / GRAPH.BD], { stroke: BLUE });
GRAPH.updateLabels();
};
GRAPH.changeSlope = function(dir) {
var prevDenom = GRAPH.MD;
GRAPH.MD = getLCM(prevDenom, INCR);
GRAPH.MN = (GRAPH.MD / prevDenom * GRAPH.MN) +
(dir * GRAPH.MD / INCR);
GRAPH.redraw();
};
GRAPH.changeIntercept = function(dir) {
var prevDenom = GRAPH.BD;
GRAPH.BD = getLCM(prevDenom, INCR)
GRAPH.BN = (GRAPH.BD / prevDenom * GRAPH.BN) +
(dir * GRAPH.BD / INCR);
GRAPH.redraw();
};
</div>
<div class="solution" data-type="custom">
<div class="instruction">
<div style="clear: left;">
Adjust this equation by clicking on the arrows:
</div>
<div class="graphie">
init({
range: [[0, 4], [0.5, 2.5]]
});
label([0, 1.5], "y = ", "right");
var slopeLabel = label([1.7, 1.5], "1", "left");
label([1.45, 1.5], "x", "right");
signLabel = label([1.8, 1.5], "+", "right");
var interceptLabel = label([2.2, 1.5], "1", "right");
addMouseLayer();
addArrowWidget({
coord: [1.4, 2.1],
direction: "up",
onClick: function() {
GRAPH.changeSlope(1);
}
});
addArrowWidget({
coord: [1.4, 0.8],
direction: "down",
onClick: function() {
GRAPH.changeSlope(-1);
}
});
addArrowWidget({
coord: [2.5, 2.1],
direction: "up",
onClick: function() {
GRAPH.changeIntercept(1);
}
});
addArrowWidget({
coord: [2.5, 0.8],
direction: "down",
onClick: function() {
GRAPH.changeIntercept(-1);
}
});
GRAPH.updateLabels = function() {
slopeLabel.remove();
signLabel.remove();
interceptLabel.remove();
slopeLabel = label([1.7, 1.5],
fractionReduce(GRAPH.MN, GRAPH.MD), "left");
signLabel = label([1.8, 1.5],
(GRAPH.BN < 0 ? "-" : "+"), "right");
interceptLabel = label([2.2, 1.5],
fractionReduce(abs(GRAPH.BN), GRAPH.BD),
"right");
};
</div>
</div>
<div class="guess">
[GRAPH.MN / GRAPH.MD, GRAPH.BN / GRAPH.BD]
</div>
<div class="validator-function">
if ((abs(guess[0] - M) < 0.001) &&
(abs(guess[1] - B) < 0.001)) {
return true;
}
if (guess[0] === 1 && guess[1] === 1) {
return "";
}
return false;
</div>
<div class="show-guess">
guess = guess.length ? guess : [1, 1, 1, 1];
GRAPH.MN = guess[0];
GRAPH.MD = 1;
GRAPH.BN = guess[1];
GRAPH.BD = 1;
GRAPH.updateLabels();
GRAPH.redraw();
</div>
</div>
</div>
</div>
</div>
</body>
</html>