Tutor profile: Manroocha S.

$$ Question\ Subject: 6th\ Grade\ Math$$ $$A\ school\ has\ 90\ pencils\ to\ give\ out\ to\ it's\ students. There\ are\ 3\ classrooms\, each\ with\ 5\ students\ in\ them. How\ many\ pencils\ will\ each\ classroom\ get? How\ many\ pencils\ will\ each\ student\ get?$$

$$ The\ first\ thing\ to\ note\ here\ is\ that\ there\ are\ two\ questions\ we\ need\ to\ answer, each\ having\ their\ unique\ solution. One\ question\ is\ asking\ how\ many\ pencils\ each\ of\ the\ 3\ classrooms\ will\ get; the\ second\ question\ is\ focused\ on\ how\ many\ pencils\ each\ student\ will\ get. $$ $$Let's\ start\ with\ how\ many\ pencils\ each\ classroom\ will\ get. There\ are\ a\ total\ of\ 90\ pencils, with\ 3\ classrooms\ to\ divide\ them\ evenly\ between. Since\ we\ are\ making\ equal\ groups, we\ use\ division\ to\ help\ us\ reach\ the\ answer. We\ can\ construct\ the\ following\ equation\ model: \frac{Total\ Units}{Number\ of\ Equal\ Groups}=Number\ of\ Units\ for\ Each\ Group$$ $$Let's\ use\ the\ numbers\ we\ have\ in\ the\ problem\ to\ solve\ the\ equation\ we\ created:$$ $$\frac{90}{3}= ?$$ $$Now, we\ can\ always\ use\ multiplication\ to\ help\ us\ fill\ in\ the\ question\ mark. Since\ multiplication\ is\ the\ opposite\ of\ division\ we\ can\ rewrite\ the\ equation\ to\ get\ the\ following: 3*?=90. Using\ either\ method, we\ get\ 30\ pencils. Therefore, the\ answer\ to\ the\ first\ question\ is\ that\ each\ classroom\ will\ get\ 30\ pencils. $$ $$Now, we\ must\ zoom\ in\ and\ figure\ out\ how\ many\ pencils\ each\ student\ will\ receive. We\ can\ use\ our\ newfound\ 30\ pencils\ per\ classroom\ to\ help\ us\ with\ that. There\ are\ a\ total\ of\ 30\ pencils\, and\ we\ must\ divide\ them\ evenly\ between\ 6\ students. Our\ equation\ model\ from\ above\ gives\ us\ a\ way\ to\ plug\ in\ these\ values: $$ $$\frac{30\ pencils}{6\ students}= ? pencils\ per\ student. $$ $$We\ can\ also\ rewrite\ this\ into\ the\ multiplication\ problem\ 6 * ? = 30. This\ leads\ us\ to\ 5, or\ 5\ pencils\ that\ each\ student\ will\ receive.$$

$$ Calculus\space is\space a\space sector\space of\space math\space that\space pertains\space heavily\space to\space analyzing\space real\space world\space scenarios. \space Take\space a\space look\space at\space the\space equation\space below: $$ $$Say\space a\space water\space tank\space is\space being\space filled\space and\space the\space volume, in\space liters, of\space the\space water\space in\space the\space tank\space after\space t\space seconds\space is\space given\space by\space the\space linear\space function: $$ $$ V^1(t )= 0.2 t^2 $$ $$ What\space would\space the\space instantaneous\space rate\space of\space change\space be\space at\space 4\space seconds? $$

$$ First, we\ need\ to\ look\ at\ the\ equation\ and\ think\ about\ what\ information\ it\ is\ providing\ us\ with. A\ preliminary\ look\ would\ allow\ us\ to\ see\ that\ we\ have\ an\ equation\ that\ represents\ how\ much\ water\ we\ have\ at\ a\ certain\ point. For\ example, if\ we\ wanted\ to\ see\ how\ much\ water\ we\ had\ at\ 4\ seconds, we\ would \set 4=t and\ plug\ that\ into\ our\ original\ equation. \The\ resulting\ answer\ would\ be\ V(4)=0.2 (4)^2, which\ would\ be\ equal\ to\ V(4) = 0.2 * 16, which\ would\ result\ in\ the\ final\ answer\ of\ V(4) = 3.2 liters. (Remember\ due\ to\ PEMDAS\ and\ our\ order\ of\ operations\, we\ would\ need\ to\ first\ square\ 4\ and\ then\ multiply\ that\ product\ by\ 0.2.) $$ $$ However\, the\ question\ wants\ us\ to\ find\ the\ instantaneous\ rate\ of\ change\ at\ a\ certain\ point\ . It's\ important\ to\ notice\ the\ difference\ between\ the\ instantaneous\ rate\ of\ change\ and\ the\ average\ rate\ of\ change. The average\ rate\ of\ change\ would\ be\ looking\ at\ the\ original\ equation, \and\ asking\ ourself\ what\ the\ rate\ of\ change\ would\ be\ across\ the\ entire\ graph, \or\ over\ a\ particular\ interval. The\ instantaneous\ rate\ of\ change\ is\ looking\ at\ what\ the\ rate\ of\ change\ is\ at\ a\ particular\ point. We\ want\ to\ note\ the\ behavior\ of\ the\ function\ at\ a\ certain\ point, \in\ \this\ case\ at\ the\ 4th\ second, \to\ see\ what\ the\ behavior\ is\ like. Think\ of\ it\ like\ taking\ a\ snapshot\ in\ a\ moment\ of\ time: \we\ want\ to\ see\ the\ rate\ of\ change\ in\ just\ that\ snapshot, \not\ the\ entire\ graph\ or\ whole\ story. $$ $$W\e use\ the\ first\ derivative\ of\ the\ original\ equation\ to\ find\ the\ instantaneous\ rate\ of\ change\ at\ a\ certain\ point. The\ derivative\ allows\ us\ to\ measure\ the\ steepness\ of\ a\ graph\ at\ a\ certain\ point\, so\ it\ provides\ us\ with\ a\ new\ equation\ that\ can\ help\ us\ take\ these\ screenshots$$. $$Here\ is\ the\ original\ equation\: V^1(t )= 0.2 t^2 $$ $$Let's\ take\ the\ derivative! V'(t)= 2 *0.2 * t ^ (2-1)$$ $$This\ is\ equal\ to\ V'(t) = 0.4 * t^1, or V'(t) = 0.4 t$$ $$If\ we\ want\ to\ find\ the\ instantaneous\ rate\ of\ change\ at\ the\ 4th\ second\, we\ plug\ in\ t=4\ into\ our\ new\ derivative. $$ $$V'(4) = 0.4 (4) = 1.6$$ $$ One\ last\ thing\ to\ note\ here\ are\ the\ units\ we\ use\ for\ the\ derivative. We\aren't\ simply\ calculating\ liters\, or\ the\ amount\ of\ water\ at\ a\ particular\ point. We\ are\ now\ calculating\ the\ rate\ of\ change\ at\ the\ 4th\ second, which\ means\ our\ unit\ has\ to\ reflect\ that. To\ create\ the\ new\ unit, we\ model\ this\ rate\ of\ change\ using\ the\ vocabulary. Rate\ of\ change= [{liters}{seconds}, so\ our\ final\ answer\ would\ be\ 1.6 liters/second. This\ translated\ would\ mean\ that\ at\ the\ 4th\ second, the\ instantaneous\ rate\ of\ change\ of\ our\ water\ that\ 1.6\ liters\ are\ added\ per\ every\ second\. $$

$$ Question\ Subject: Understanding\ ACT\ Math\ Vocabulary$$ $$ Question: The\ ACT\ Math\ section\ is\ known\ for\ centering\ around\ certain\ subject-specific\ vocabulary.Do\ your\ best\ to\ write\ the\ definition\ and\ an\ example\ of\ the\ following\ words\ to\ the\ best\ of\ your\ ability:$$ $$integer, equivalent, reciprocals, expression, equation$$

$$integer: a\ whole\ number\, a\ number\ that\ is\ NOT\ a\ fraction. i.e: 6$$ $$equivalent: two\ expressions\ that\ are\ equal\ in\ value\ to\ each\ other. i.e: 5+5=8+2 $$ $$reciprocals: two\ numbers\ (A\ and\ B)\ that\ when\ multiplied, their\ product\ is\ equal\ to\ 1. i.e,: 1/2\ and\ 2 $$ $$ expression: Numbers, symbols, or operations (+,-, etc)\ grouped\ together\ to\ show\ the\ value\ of\ something\ (i.e: 7+5, 2 x 3) $$ $$equation: a\ statement\ that\ the\ value\ of\ two\ mathematical\ expressions\ are\ equal. i.e: 4 + 5 = 9 $$