(:title Slope of a Line:) >>nav_bar<< %yel%[[glossary/slopemc|Mathcasts]]         %exa%Examples      %newwin%[[http://mathcasts.org/gg/enliven/af/lines/slopes/slopes_index.html|%pra%Practice%%]]    %mor%More      %up%[[glossary/S]] >><< [[<<]] [table border=1 cellpadding=3 width=830] [row] []>>&<< %ref b%Definition:%% The %red b%slope%% is the %red b%rate of change%% of a line, that is the: %c11 bgr%"vertical change"%c11 dkr%/%c11 bgr%%c11 up%"horizontal change"%c11 dkr%. [row] [] (:showhide init=hide div=div8 lshow="+" lhide="-":) %exa%%b%Introduction to Slopes%% - %pra%See, Hear & Do%% - Money in the Bank %% %red b%NEW!%% (:div8 id=div8 :) (:div style="margin-left:20px":) %popwin width=600 height=450%[[http://mathcasts.org/gg/enliven/af/lines/slopes/Slope_mc1/Slope_mc1.htm| %yel%Introduction to Slopes 1.1 - Mathcast: Money in the Bank%%]] -> [[http://mathcasts.org/gg/enliven/af/lines/slopes/bank-1.html| %pra%Money in the Bank: Worksheet%%]] %popwin width=600 height=450%[[http://mathcasts.org/gg/enliven/af/lines/slopes/Slope_mc2/Slope_mc2.htm| %yel%Introduction to Slopes 1.2 - Mathcast: More Positive%%]] %popwin width=600 height=450%[[http://mathcasts.org/gg/enliven/af/lines/slopes/Slope_mc3/Slope_mc3.htm| %yel%Introduction to Slopes 1.3 - Mathcast: Slope & Speed%%]] -> %popwin width=950 height=680%[[http://mathcasts.org/gg/enliven/af/lines/slopes/car_race_anim.html| %up%Slope & Speed - InterActivity%%]] %red b%NEW!%% %popwin width=600 height=450%[[http://mathcasts.org/gg/enliven/af/lines/slopes/Slope_mc4/Slope_mc4.htm| %yel%Introduction to Slopes 1.4 - Mathcast: Formula for Slope%%]] %popwin width=600 height=450%[[http://mathcasts.org/gg/enliven/af/lines/slopes/Slope_mc5/Slope_mc5.htm| %yel%Introduction to Slopes 1.5 - Mathcast: Calculating Slope%%]] (:divend:) (:div8end :) [tableend] >>-<< [table border=1 cellpadding=3 width=830] [row] [c width=310] {gg width=300; height=220; xmin=-5; xmax=10; xscl=1; ymin=-8; ymax=3; yscl=1; axes(); fontfamily="sansserif"; fontsize="10"; stroke="#009900"; strokewidth=3; plot("2*x-6"); strokewidth=2; stroke="#990099"; strokedasharray="5,5"; endpoints=" ->";line([1,-4],[4,-4]); d=text([4,-1.5], "change in y = +6",right); stroke="#009999";line([4,-4],[4,2]);text([1,-4.5],"change in x = +3",right);stroke="#000000"; circle([1,-4],.1); circle([4,2],.1); fontweight="bold"; text([1,-4],"A",aboveleft);text([4,2],"B",aboveleft); gg} [] %pra%To calculate the slope we "pick" points on the line A(%up%1%pra%,%bgr%-4%pra%) and B(%up%4%%,%bgr%2%pra%). >>&<>&<< [table border=1 cellpadding=3] [row] []%up%The "change in x" "from A to B" is 4-1=3.%% >>&<<%bgr%The "change in y" "from A to B" is 2-(-4)=2+4=6.%% [tableend] >>&<< [table border=2 cellpadding=3] [row] []%c11%Slope of this line = m = %% {$\frac{\mbox{vertical change}}{\mbox{horizontal change}} = \frac{6}{3} = 2$}%% [tableend] >>-<< The equation of this line is: {$y=2x-6$} or {$2x-y=6$}. (:div911 class=s9 :) >>&<< (:showhide init=hide div=div912 lshow="+" lhide="-":) %exa%What if we "pick" different points?%% >>-<< [row] [ colspan=2] (:div912 id=div912 style="margin-left:20px" :) [table border=1 cellpadding=2] [row] [c width=306] {gg width=300; height=220; xmin=-5; xmax=10; xscl=1; ymin=-8; ymax=3; yscl=1; axes(); fontfamily="sansserif"; fontsize="10"; stroke="#009900"; strokewidth=3; plot("2*x-6"); strokewidth=2; stroke="#660000"; circle([3,0],.1); circle([3,0],.2); b=text([3,0],"root",aboveleft); stroke="#660099"; circle([0,-6],.2); c=text([0,-6],"y-intercept",belowright);stroke="#990099"; strokedasharray="5,5"; endpoints="*->";line([3,0],[0,0]); d=text([4,-1.5], "change in y = -6",right); stroke="#009999";line([0,0],[0,-6]);text([1,-4.5],"change in x = -3",right); gg} []This is the same line as above. %pra%Let's "pick" the points%% root=(%up%3%%,%bgr%0%%) and y-intercept=(%up%0%%,%bgr%-6%%).%% >>&<<     Remember - we subtract "second point" - "first point". >>&<< [table border=1 cellpadding=3] [row] []%up%"change in x" = 0-3 = -3%% and %bgr%"change in y" = -6-0 = -6%% [tableend] >>&<< [table border=1 cellpadding=3] [row] []%c11%Slope = m = %% {$\frac{\mbox{vertical change}}{\mbox{horizontal change}} = \frac{-6}{-3} = 2$}. [tableend] >>-<< [table border=2 bordercolor=#990000 cellpadding=3] [row] []%exa b%''Slope is same no matter what points we "pick"''.%% [tableend] >>&<< '''BUT''' you must do the subtractions in the same direction! (:divend:) [tableend] (:div912end:) >>&<< (:div911end:) [row] [ colspan=2] (:showhide init=hide div=div21 lshow="+" lhide="-":) Which comes first: %c11 up%x%% or %c11 bgr%y%%? (:div21 id=div21 :) [table border=1 cellpadding=3] [row] [ colspan=2] * %c11 exa%Coordinates of a point are (x,y)%% so when %c11 exa%graphing a point%% do %c11 up%horizontal%% then %c11 bgr%vertical%%. [[http://mathcasts.org/gg/enliven/af/coor/coor1/coor1-1.html|See&Do]] [row] [ colspan=2] * %c11 reg%Slopes%% are  %c11%{$\frac{\mbox{vertical change}}{\mbox{horizontal change}}$}%%  - so with %c11 reg%slope%% the %c11 bgr%vertical change%% is above the %c11 up%horizontal change%% in the fraction! >>&<< [row] [ colspan=2] * Think: slope=speed. Speed is measured distance/time (mph or m/s) where %up%time = x-axis%% and %bgr%distance = y-axis%%. [tableend] (:div21end :) [tableend] >>&<< [table border=1 bordercolor=#ee8800 cellspacing=3 width=830] [row] [] %reg b%Calculating the slope of a line:%% # Pick any 2 points on the line. # Arbitrarily decide which is the "first point" and "second point". # "change in x" = "x-coordinate of %b%second point%%" - "x-coordinate of %b%first point%%". # "change in y" = "y-coordinate of %b%second point%%" - "y-coordinate of %b%first point%%". # %c11%slope =  %% {$\frac{\mbox{change in y}}{\mbox{change in x}}$} [row] [] %reg b%Rule:%% The slope of a line is a number. Parallel lines have the same slope. [row] [] %pra b%InterActivities:%% %newwin%[[http://mathcasts.org/gg/enliven/af/lines/slopes/slopes_index.html|%pra%5 InterActivities to Practice Slope%%]] [tableend] >>-<< ---- >>&<< %rel%Related Topics *[[Lines]] *[[GlossaryT/Systems2|2x2 Linear Systems - Two Lines in the Plane]] *[[GlossaryT/LinearEquation|Linear Equation - Not always a line]] *[[GlossaryT/LinearFunction|Linear Function - Always a line]] *[[InterA/Cartcoor2|Cartesian Coordinates 2 - Working with Points in the Plane]] - Review *[[InterA/Cartcoor1|Cartesian Coordinates 1 - Plotting and Identifying Points in the Plane]] - Review >>&<< ---- >>&<< (:showhide init=hide div=div9 lshow="+" lhide="-":) Metadata (:div9 id=div9 :) [table class=column border=1 width=100% cellpadding=3] [row] [ width=110px] '''Global''' []Slope of a line [row] []Brief []Mathcasts and Interactivities to understand and practice the slope of a line in the Cartesaian plane. [row] []Grade []7-10    Interactivities start at 7th grade level on up [row] []Strand []Algebra and Functions / Geometry and Measurement [row] []Standard []%popwin%[[standards/CA7AF-3-3?action=popopen|CA 7AF.3.3]], %popwin%[[standards/Is1al2-5?action=popopen|Algebra 1 2.5]], %popwin%[[standards/ACTgr20?action=popopen|ACT GR 20-23]], %popwin%[[standards/ACTgr24?action=popopen|ACT GR 24-27]] [row] []Keywords []line, slope, linear equation, change in y, change in x, ratio [row] []Comments []none [row] []Download []  [row] []Author []LFS - [[mailto:emath@emathforall.com?subject=slopes|contact]] - [[http://math247.pbwiki.com/Algebra-with-GeoGebra|website]] [row] []Type []Freeware - Available for [[http://mathcasts.org/gg/student/lines/slopes/slopes.zip|School]] and [[http://mathcasts.org/gg/student/lines/slopes/slopes_offline.zip|Offline]] and Online Use - Translatable (html) [row] []Use []Requires [[http://www.java.com/en/download/index.jsp|sunJava player]] [tableend] (:div9end:) >>&<< [table width=100%] [row] [][[Glossary/S| Attach:main/tri_purple_up_a.gif ]] [[Glossary/S| Up one level]] [r](:html:) (:htmlend:) [tableend]