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volume_with_unit_cubes.html
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volume_with_unit_cubes.html
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<!DOCTYPE html>
<html data-require="math graphie graphie-3d math-format interactive word-problems">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/>
<title>Volume with unit cubes</title>
<script src="../khan-exercise.js"></script>
<script type="text/javascript">
var makeSquare = function(left, width, bottom, height, color) {
KhanUtil.currentGraph.path([
[left, bottom],
[left, bottom + height],
[left + width, bottom + height],
[left + width, bottom],
true
],
{ stroke: "black", strokeWidth: 1, fill: color }
);
};
var normalize = function(vect) {
var length = Math.sqrt(vect[0] * vect[0] +
vect[1] * vect[1] +
vect[2] * vect[2]);
return _.map(vect, function(e) { return e / length; });
};
</script>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="UNIT_TEXT,UNIT">randFromArray([
["\\text{ " + $._("cm") + "}", new Plural(function(num) {
return $.ngettext("centimeter", "centimeters", num);
})],
["\\text{ " + $._("m") + "}", new Plural(function(num) {
return $.ngettext("meter", "meters", num);
})],
["\\text{ " + $._("in") + "}", new Plural(function(num) {
return $.ngettext("inch", "inches", num);
})],
["\\text{ " + $._("ft") + "}", new Plural(function(num) {
return $.ngettext("foot", "feet", num);
})]
])</var>
</div>
<div class="problems">
<div id="find-volume-count">
<div class="vars" data-ensure="DIMENSIONS[0]*DIMENSIONS[1]*DIMENSIONS[2]<33&&DIMENSIONS[0]*DIMENSIONS[1]*DIMENSIONS[2]>1">
<var id="DIMENSIONS" >randRange(1,4,3)</var>
<var id="LENGTH">DIMENSIONS[0]</var>
<var id="WIDTH">DIMENSIONS[1]</var>
<var id="HEIGHT">DIMENSIONS[2]</var>
<var id="MAX">max(max(LENGTH, WIDTH), HEIGHT)</var>
<var id="XBORD">LENGTH/MAX</var>
<var id="YBORD">WIDTH/MAX</var>
<var id="ZBORD">HEIGHT/MAX</var>
<var id="COLOR1">"#D7ED3A"</var>
<var id="COLOR2">"#38C77F"</var>
<var id="COLOR3">"#78D0EF"</var>
<var id="COLOR4">"#F0B63A"</var>
<var id="VERTICES">[
[LENGTH/MAX, WIDTH/MAX, HEIGHT/MAX],
[LENGTH/MAX, WIDTH/MAX, -HEIGHT/MAX],
[LENGTH/MAX, -WIDTH/MAX, -HEIGHT/MAX],
[LENGTH/MAX, -WIDTH/MAX, HEIGHT/MAX],
[-LENGTH/MAX, WIDTH/MAX, HEIGHT/MAX],
[-LENGTH/MAX, WIDTH/MAX, -HEIGHT/MAX],
[-LENGTH/MAX, -WIDTH/MAX, -HEIGHT/MAX],
[-LENGTH/MAX, -WIDTH/MAX, HEIGHT/MAX]
]</var>
<var id="SCALE">4.0</var>
<var id="FACES">[
{
verts: [3, 2, 1, 0],
color: COLOR1
}, {
verts: [0, 1, 5, 4],
color: COLOR3
}, {
verts: [3, 0, 4, 7],
color: COLOR2
}
]</var>
</div>
<p class="question">
What is the volume of this box?
</p>
<p>
Note: Each cube is a cubic <var>UNIT</var>
</p>
<div class="graphie" id = "box">
init({range: [[-2, 2], [-2, 2]], scale: [100, 100]});
addMouseLayer();
var start1 = 7;
for (var i = 1; i < LENGTH;i++){
VERTICES.push([(-LENGTH+2*i)/MAX,-WIDTH/MAX,HEIGHT/MAX]);
};
for (var i = 1; i < LENGTH;i++){
VERTICES.push([(-LENGTH+2*i)/MAX,WIDTH/MAX,HEIGHT/MAX]);
};
for (var i = 1; i < LENGTH;i++){
VERTICES.push([(-LENGTH+2*i)/MAX,WIDTH/MAX,-HEIGHT/MAX]);
};
var start2=VERTICES.length-1;
for (var i = 1; i < WIDTH;i++){
VERTICES.push([-LENGTH/MAX,(-WIDTH+2*i)/MAX,HEIGHT/MAX]);
};
for (var i = 1; i < WIDTH;i++){
VERTICES.push([LENGTH/MAX,(-WIDTH+2*i)/MAX,HEIGHT/MAX]);
};
for (var i = 1; i < WIDTH;i++){
VERTICES.push([LENGTH/MAX,(-WIDTH+2*i)/MAX,-HEIGHT/MAX]);
};
var start3 = VERTICES.length-1;
for (var i = 1; i < HEIGHT;i++){
VERTICES.push([LENGTH/MAX,-WIDTH/MAX,(-HEIGHT+2*i)/MAX]);
};
for (var i = 1; i < HEIGHT;i++){
VERTICES.push([LENGTH/MAX,WIDTH/MAX,(-HEIGHT+2*i)/MAX]);
};
for (var i = 1; i < HEIGHT;i++){
VERTICES.push([-LENGTH/MAX,WIDTH/MAX,(-HEIGHT+2*i)/MAX]);
};
//graphing the initial object
graph.obj = make3dObject(VERTICES, { scale: SCALE });
_.each(FACES, function(face) {
graph.obj.addFace(face);
});
for (var i = 1;i<LENGTH;i++){
graph.obj.addSketch({ verts: [start1+i,start1+i+LENGTH-1,start1+i+2*LENGTH-2], opacityValue:0.5});
};
for (var i = 1;i<WIDTH;i++){
graph.obj.addSketch({ verts: [start2+i,start2+i+WIDTH-1,start2+i+2*WIDTH-2], opacityValue:0.5});
};
for (var i = 1;i<HEIGHT;i++){
graph.obj.addSketch({ verts: [start3+i,start3+i+HEIGHT-1,start3+i+2*HEIGHT-2], opacityValue:0.5});
};
graph.obj.setPos([0, 0, 5]);
graph.obj.rotate(1,0 , 0, PI/3);
graph.obj.rotate(0, 1, 0, 0);
graph.obj.rotate(0, 0, 1, PI/6);
graph.obj.doDraw();
GRAPH = graph;
</div>
<div class="solution" data-type="multiple">
<p><span class="sol" data-type="rational"><var>LENGTH*WIDTH*HEIGHT</var></span> <code><var>UNIT_TEXT</var>^3</code></p>
</div>
<div class="hints">
<p>
The volume is measured by counting the number of cubic <var>plural_form(UNIT)</var>.
</p>
<p>
Carefully count the cubes. Some of the cubes might be hidden behind other cubes. Try to visualize all of the cubes.
</p>
<p>
There are a total of <code><var>LENGTH*WIDTH*HEIGHT</var></code> cubes.
</p>
</div>
</div>
<div id="weirdShape" >
<div class="vars">
<var id=VOL>randRange(3,8)</var>
<var id="COLOR1">"#D7ED3A"</var>
<var id="COLOR2">"#38C77F"</var>
<var id="COLOR3">"#78D0EF"</var>
<var id="COLOR4">"#F0B63A"</var>
<var id="VERTICES">[]</var>
<var id="SCALE">2.0</var>
<var id="FACES">[]</var>
<var id="surfaceArea">0</var>
</div>
<p class="question">
What is the volume of this shape?
</p>
<p>
Each cube has side lengths of one <var>UNIT</var>. You can click and drag to rotate the shape.
</p>
<div class="graphie" id = "box">
init({range: [[-2, 2], [-2, 2]], scale: [100, 100]});
addMouseLayer();
//cubes stores the positions of the cubes
var cubes = [[0,0,0]];
// inArray checks whether a point is in the array already.
// For example inArray([0,0,0],[[0,0,0,],[0,0,1]]) = true,
// but inArray([0,0,2],[[0,0,0],[0,0,1]]) = false. This is
// only used for an ordered triple and an array of ordered
// triples.
var inArray = function(point,array) {
for (var i=0; i< array.length;i++) {
if (point[0] == array[i][0] && point[1] == array[i][1] && point[2] == array[i][2]){
return true;
}
}
return false;
}
//This will store the boundary of the current set of cubes
//Add cubes to cubes from the current boundary until we have VOL cubes
for (var i = 0; i < VOL-1; i++){
var boundary = [];
// Figuring out the current boundary
for (var j = 0; j<cubes.length;j++){
var cube0 = [cubes[j][0]+1,cubes[j][1],cubes[j][2]];
var cube1 = [cubes[j][0],cubes[j][1]+1,cubes[j][2]];
var cube2 = [cubes[j][0],cubes[j][1],cubes[j][2]+1];
var cube3 = [cubes[j][0]-1,cubes[j][1],cubes[j][2]];
var cube4 = [cubes[j][0],cubes[j][1]-1,cubes[j][2]];
var cube5 = [cubes[j][0],cubes[j][1],cubes[j][2]-1];
if (!inArray(cube0,cubes) && ! inArray(cube0,boundary)){
boundary.push(cube0);
}
if (!inArray(cube1,cubes) && ! inArray(cube1,boundary)){
boundary.push(cube1);
}
if (!inArray(cube2,cubes) && ! inArray(cube2,boundary)){
boundary.push(cube2);
}
if (!inArray(cube2,cubes) && ! inArray(cube2,boundary)){
boundary.push(cube2);
}
if (!inArray(cube3,cubes) && ! inArray(cube3,boundary)){
boundary.push(cube3);
}
if (!inArray(cube4,cubes) && ! inArray(cube4,boundary)){
boundary.push(cube4);
}
}
cubes.push(boundary[randRange(0,boundary.length-1)]);
}
// Addind vertices and faces
for (var i = 0; i < VOL; i++) {
var index = VERTICES.length;
VERTICES.push([cubes[i][0]+0.5,cubes[i][1]+0.5,cubes[i][2]-0.5]);
VERTICES.push([cubes[i][0]+0.5,cubes[i][1]-0.5,cubes[i][2]-0.5]);
VERTICES.push([cubes[i][0]-0.5,cubes[i][1]-0.5,cubes[i][2]-0.5]);
VERTICES.push([cubes[i][0]-0.5,cubes[i][1]+0.5,cubes[i][2]-0.5]);
VERTICES.push([cubes[i][0]+0.5,cubes[i][1]+0.5,cubes[i][2]+0.5]);
VERTICES.push([cubes[i][0]+0.5,cubes[i][1]-0.5,cubes[i][2]+0.5]);
VERTICES.push([cubes[i][0]-0.5,cubes[i][1]-0.5,cubes[i][2]+0.5]);
VERTICES.push([cubes[i][0]-0.5,cubes[i][1]+0.5,cubes[i][2]+0.5]);
//only add a face if there is not another cube next to it
if (!inArray([cubes[i][0]+1,cubes[i][1],cubes[i][2]],cubes)) {
FACES.push({
verts: [index,index+4,index+5,index+1],
color: COLOR1
});
surfaceArea++;
}
if (!inArray([cubes[i][0]-1,cubes[i][1],cubes[i][2]],cubes)) {
FACES.push({
verts: [index+7,index+3,index+2,index+6],
color: COLOR1
});
surfaceArea++;
}
if (!inArray([cubes[i][0],cubes[i][1]+1,cubes[i][2]],cubes)) {
FACES.push({
verts: [index,index+3,index+7,index+4],
color: COLOR2
});
surfaceArea++;
}
if (!inArray([cubes[i][0],cubes[i][1]-1,cubes[i][2]],cubes)) {
FACES.push({
verts: [index+5,index+6,index+2,index+1],
color: COLOR2
});
surfaceArea++;
}
if (!inArray([cubes[i][0],cubes[i][1],cubes[i][2]+1],cubes)) {
FACES.push({
verts: [index+4,index+7,index+6,index+5],
color: COLOR3
});
surfaceArea++;
}
if (!inArray([cubes[i][0],cubes[i][1],cubes[i][2]-1],cubes)) {
FACES.push({
verts: [index+3,index,index+1,index+2],
color: COLOR3
});
surfaceArea++;
}
}
// graphing the initial object
graph.obj = make3dObject(VERTICES, { scale: SCALE, facesTransparent: false, faceBorder:true });
_.each(FACES, function(face) {
graph.obj.addFace(face);
});
graph.obj.setPos([0, 0, 5]);
graph.obj.rotate(1,0 , 0, PI/3);
graph.obj.rotate(0, 1, 0, 0);
graph.obj.rotate(0, 0, 1, PI/8);
graph.obj.doDraw();
GRAPH = graph;
var mouseTarget = mouselayer.rect(0, 0, 400, 400).attr({
fill: "#000", opacity: 0.0
});
$(mouseTarget[0]).bind("vmousedown", function(event) {
event.preventDefault();
GRAPH.lastX = event.pageX;
GRAPH.lastY = event.pageY;
$(document).bind("vmousemove vmouseup", function(event) {
event.preventDefault();
var dx = (event.pageX - GRAPH.lastX) / 200 * PI;
var dy = (event.pageY - GRAPH.lastY) / 200 * PI;
GRAPH.lastX = event.pageX;
GRAPH.lastY = event.pageY;
var xAxis = normalize(GRAPH.obj.perspective[1]);
var yAxis = normalize(GRAPH.obj.perspective[0]);
GRAPH.obj.rotate(xAxis[0], xAxis[1], xAxis[2], -dx / 2);
GRAPH.obj.rotate(yAxis[0], yAxis[1], yAxis[2], -dy / 2);
GRAPH.obj.doDraw();
if (event.type === "vmouseup") {
$(document).unbind("vmousemove vmouseup");
}
})
});
</div>
<div class="solution" data-type="multiple">
<p><span class="sol" data-type="rational"><var>VOL</var></span> <code><var>UNIT_TEXT</var>^3</code></p>
</div>
<div class="hints">
<p>
The volume of a shape can measured by the number of cubic <var>plural_form(UNIT)</var> which make up the shape.
</p>
<p>
There are <code><var>VOL</var></code> cubes.
</p>
<p>
The volume is <code><var>VOL</var> <var>UNIT_TEXT</var>^3</code>
</p>
</div>
</div>
</div>
</div>
</body>
</html>