Road to U‐Su
ubstitution Gaame (Caalculus) Learning ggoal: To begin
n to “see” thee u and du substitutions in
n problems an
nd to recognizze compositio
ons for what tthey are. I recommend inttroducing thee game in two
o phases, because it’s difficult for studeents to see how
w to structure u‐substitutiion problemss at first. This is (ideeally) a 3‐playyer game, although it can be modified ffor two people. Each grou
up of players w
will need a gaame board, a set of playerr road pieces (the green, yeellow, and blu
ue pieces),th
hree Integral C
Construction Platforms (on
ne for each player) and a d
deck of cards (rectangular grey or white ccards). Print aall game piecces on cardsto
ock for best reesults. Phase I: G
Guided Play Give each
h set of playerr a game boarrd and an Inteegral construction board aand that is all. On a docum
ment camera, laay out 8 cards so that everryone in the cclass can see them and theen demonstraate how to create one of thee integrals on
n the game bo
oard. To creaate an integraal, you need aa u, a du, and some functio
on or combination of functio
ons to put theem in (unless,, of course, yo
ou’re using th
he integral of udu). uct Example: The followingg could be used to constru
(
)
∫(x
2
+ 2 ) 2x dx = ∫ u 3 du . 3
x 2 + 3 3
2xx ⌠ sec 2 x
du
d =⌠
dx
ng four tiles ccould be used
d to constructt ⎮
Example:: The followin
⎮ 2 . 2
⌡u
⌡ ( tan x )
(
)
2
(
(
)
)
taan x sec 2 x Walk the learners thro
ough one or tw
wo examples using the document cameera. Every tim
me you constrruct an integraal, place a roaad piece on th
he game boarrd on one of tthe edges of tthe appropriaate hexagon, and then replaace the missin
ng cards. Now, lay o
out a new sett of cards for the learners to try constru
ucting integraals (make sure to lay out 8
8 cards thatt have possibilities). Go th
hrough a coup
ple rounds, w
with the “game” being the construction of as many inteegrals as posssible OR usingg as many carrds at once ass you can in one turn. Now the sstudents are ready to playy on their own
n! Let the reaal game begin
n! Created b
by Maria H. An
ndersen, TeacchingCollegeM
Math.com Phase II: Group Play To Win: B
Build the longe
est continuou
us road on the board. Witth each integrral you constrruct, you have the opportunity to build a single piece o
of the road on the edge off the appropriate hexagon. Rules: 1. Everyy player is deaalt 8 cards. Laay them out sso that one ro
ow of 4 is abo
ove the Integrral Construction Platfo
orm and one rrow of 4 is beelow the Integgral Construction Platform
m. Anything th
hat is above tthe card is fair game to
o be stolen from another p
player (more on this later)), so keep you
ur most valuable cards below or on the constructtion platform
m. You must always keep 8 cards out and 4 cards abo
ove the pllatform. No m
more, no less.. As cards aree played, theyy should be reeplaced from the deck. 2. On yo
our turn, you can do two th
hings. Option
n 1: Ideally, yyou construct an integral u
using a selection of game pieces, and p
place a road p
piece on the b
board. I thinkk it is best if sstudents writee out the inteegral paper to present to their o
opponents (ottherwise it’s jjust a they aare constructing on a sheeet of scratch p
little ttoo easy to le
et one slide byy). Option 2:: If you can’t construct anyy integrals, yo
ou will have tto sacrifice two cardss to the otherr two players (one to each)). What they take is their choice, first ccome hey can only cchoose from tthe cards abo
ove the Consttruction. You can then rep
place first served (but th
he deck, and each of the o
other players must get rid of a card (theey these two missing cards from th
discard the on
ne they just stole from you
u if they wish). may d
3. When
n a hexagon o
on the board has been surrrounded by ro
oads, you can
n remove all ggrey function cards that apply on
nly to that heexagon from tthe game. The gamee ends when tthe board is filled or when
n there is an in
ndisputable w
winner no maatter what else happens. Two‐player modificatiion: If a playeer can’t construct any integgrals, then the opposing player steals tw
wo cards (instead of one). Other mo
odifications: C
Cut out the heexagons and place them in
n some otherr arrangement (I’d suggestt making a copy to creatte a new fixed
d gameboard or there are too many mo
oving pieces). Created b
by Maria H. An
ndersen, TeacchingCollegeM
Math.com e
(
(
)
)
sec 2 (
(
(
(
(
2
e
)
)
3
)
)
2
sec
s 2(
(
e
) (
) (
(
)
)
3
(
(
)
)
2
seec 2 (
(
) (
) (
(
)
)
Created b
by Maria H. An
ndersen, TeacchingCollegeM
Math.com 3
) (
) (
x3 3x 2 )
2
(
x2 2x
x 2 + 3 2x
) (
) (
)
2
)
)
(
sin (
cos (
)
)
(
)
sin (
cos (
)
)
(
)
siin (
coos (
)
)
)
tann x sec 2 x sinn x cos x coos x − sin x ex ln x 3
e3x
2x 3ee3x 2 x ln 2 seec x tan x ex sec x Created b
by Maria H. An
ndersen, TeacchingCollegeM
Math.com x −1
x 3 + 1 6x 6x x3 3x 2 tann x sin x cos x x 2 + 3 x2
cos x − sin x
2 2x
2x
3
e3x
2x ex ln x 3ee3x 2 x ln 2 seec x tan x ex secc 2 x sec x Created b
by Maria H. An
ndersen, TeacchingCollegeM
Math.com x −1
Construction Platforms (on
ne for each player) Integral C
∫
dx
∫
dx
∫
dx
Created b
by Maria H. An
ndersen, TeacchingCollegeM
Math.com Road to U‐substitution
Game Board
⌠ du
⎮ 2
⌡u
∫ cos u du
d
∫ u du
⌠ du
⎮
⌡ u
e
du
∫
u
∫u
∫ sec
2
d
∫ u du
3
1/2
du
∫ u du
2
u du
∫ sin u du
Created by Maria H. Andersen
TeachingCollegeMath.com
Player Road Pieces
(each player is a diff
different color of road)
t l
f
d)
Created by Maria H. Andersen
TeachingCollegeMath.com