Ratio!and!Proportion!Strategies
Page |1
Comparing!Rates!
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The!problem!below!is!from!the!Connected!Mathematics!Project!(CMP2)!Comparing!and!Scaling!(Grade!7).!You!
can!find!a!similar!problem!and!work!at!www.figurethis.org!(Challenge!#25)!
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Ratio!and!Proportion!Strategies
Similar!Problem!from!CMP!1.!Note!student!work!matches!this!problem.!
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Part"Part!Ratios!
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Part"Whole!Ratios!
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Ratio!and!Proportion!Strategies
Missing!Value!Problems!(Proportions)!–!Student!Strategies!
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Wes!Problem!#1!
Two!baseballs!cost!$5.!How!much!will!seven!baseballs!cost?!
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The!unit!rate!approach!is!characterized!when!students!find!the!constant!factor!across!or!between!the!two!
measure!spaces.!!
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M1:!______________!
M2:!____________!
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The!scale!factor!approach!is!characterized!when!students!find!the!multiplicative!relationship!among!elements!
within!the!measure!space.!
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M1:!______________!
M2:!____________!
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The!table!strategy!is!a!strategy!that!is!a!mix!of!unit!
rate!and!factor!of!change.!
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The!graphing!strategy!has!an!interesting!equation.!
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The!fraction!strategy!is!devoid!of!the!problem!context.!
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Cross"multiplication!is!also!a!strategy!but!is!hard!for!students!to!explain!why!it!works!and!they!tend!not!to!use!
it!when!solving!problems!on!their!own.!
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Ratio!and!Proportion!Strategies
Page |4
Wes!Problem!#2!
I!can!buy!6!baseballs!for!$20.!How!much!will!18!baseballs!cost?!
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Wes!Problem!#3!
I!can!buy!6!baseballs!for!$18.!How!much!will!11!baseballs!cost?!
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Wes!Problem!#4!
I!can!buy!7!baseballs!for!$12.!How!much!will!11!baseballs!cost?!
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The!Marissa!Problem!(Adapted!from!Post)!
Marissa!bought!0.46!of!a!pound!of!wheat!flour!for!which!she!paid!$0.83.!How!many!pounds!could!she!buy!for!
one!dollar?!
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The!Track!Problem!(Adapted!from!Cramer)!
Wendy!and!Brooke!run!at!the!same!speed!around!a!track.!Wendy!was!running!lap!3!when!Brooke!was!running!
lap!7.!What!lap!will!Brooke!be!on!when!Wendy!is!running!lap!9?!
Page |5
Ratio!and!Proportion!Strategies
Solve!the!Marissa!Problem!using!each!of!the!strategies!written!below.!
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The!Marissa!Problem!
Marissa!bought!0.46!of!a!pound!of!wheat!flour!for!which!she!paid!$0.83.!How!
many!pounds!could!she!buy!for!one!dollar?!
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The!unit!rate!approach!(between!measures)!!
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M1:!______________!
M2:!____________!
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The!scale!factor!approach!(within!measures)!
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M1:!______________!
M2:!____________!
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The!table!strategy!is!a!strategy!
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The!graphing!strategy!with!equation!
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The!fraction!strategy!
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Cross"multiplication!
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Ratio!and!Proportion!Strategies
Page |6
Proportional!Situation!or!Not?!
Answer!each!problem!below.!Decide!if!the!situation!is!proportional!or!not.!If!the!situation!is!proportional!
then!find!a!rule!of!the!form!y!=!mx!that!describes!the!relationship.!
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Taxi!cabs!in!Minneapolis!charge!$2.50!for!the!first!
one"fifth!mile!and!each!succeeding!fifth!will!cost!
$.38.!How!much!will!the!fare!be!for!a!2!mile!trip?!
The!action!figure!of!a!famous!wrestler!has!a!scale!
factor!of!1/12.!If!the!doll!is!7!inches!tall,!how!tall!is!
the!actual!wrestler?!
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Dara!swims!four!laps!across!the!pool!in!the!same!
time!as!Michael!swims!three!laps.!They!begin!
swimming!at!the!same!time.!How!far!has!Michael!
swum!when!Dara!completes!lap!10?!
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Jake!typically!exchanges!200!text!messages!a!day.!
The!plan!he!signed!up!for!charges!a!$30!flat!fee!
per!month!and!does!not!charge!for!the!first!1000!
text!messages!exchanged.!Each!message!
exchanged!after!the!first!1000!costs!$0.05.!What!
is!the!amount!for!Jake’s!bill!for!the!month!of!
October?!
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Two!U.S.!dollars!can!be!exchanged!for!three!
Euros.!How!many!U.S.!dollars!can!be!exchanged!
for!21!Euros?!
Lance!is!riding!his!bicycle!in!France.!He!rides!18!
km!in!half!an!hour.!If!he!rides!at!the!same!rate,!
how!long!will!it!take!him!to!ride!162!km?!
Ratio!and!Proportion!Strategies
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Page |7
Write!a!missing!value!problem!that!would!encourage!a!unit!rate!strategy.!
Write!as!missing!value!problem!that!would!encourage!a!scale!factor!strategy.!!
Write!a!missing!value!problem!that!has!a!non"integer!factor!between!and!within!measures.!
Write!a!missing!value!problem!that!is!not!a!proportion.!You!would!use!this!type!of!problem!if!you!were!
interested!in!proportional!reasoning!in!students!that!may!know!how!to!solve!proportions!(possibly!
using!cross"multiplication)!but!always!apply!this!method!when!they!know!three!values!and!are!missing!
the!fourth.!
Write!a!missing!value!problem!that!would!encourage!multiplicative!strategies!and!make!correct!
arithmetic!strategies!difficult.!
Ratio!and!Proportion!Strategies
Page |8
2007!Minnesota!Algebra!Standards!(http://education.state.mn.us)!
Grade!5!
1. Recognize!and!represent!patterns!of!change;!use!patterns,!tables,!graphs!and!rules!to!solve!real"world!and!
mathematical!problems.!
2. Use!properties!of!arithmetic!to!generate!equivalent!numerical!expressions!and!evaluate!expressions!
involving!whole!numbers.!
3. Understand!and!interpret!equations!and!inequalities!involving!variables!and!whole!numbers,!and!use!them!
to!represent!and!solve!real"world!and!mathematical!problems.!
Grade!6!
4. Recognize!and!represent!relationships!between!varying!quantities;!translate!from!one!
representation!to!another;!use!patterns,!tables,!graphs!and!rules!to!solve!real"world!and!
mathematical!problems.!
5. Use!properties!of!arithmetic!to!generate!equivalent!numerical!expressions!and!evaluate!
expressions!involving!positive!rational!numbers.!
6. Understand!and!interpret!equations!and!inequalities!involving!variables!and!positive!rational!
numbers.!Use!equations!and!inequalities!to!represent!real"world!and!mathematical!problems;!use!
the!idea!of!maintaining!equality!to!solve!equations.!Interpret!solutions!in!the!original!context.!
Grade!7!
7. Understand!the!concept!of!proportionality!in!real"world!and!mathematical!situations,!and!
distinguish!between!proportional!and!other!relationships.!
8. Recognize!proportional!relationships!in!real"world!and!mathematical!situations;!represent!these!
and!other!relationships!with!tables,!verbal!descriptions,!symbols!and!graphs;!solve!problems!
involving!proportional!relationships!and!explain!results!in!the!original!context.!
9. Apply!understanding!of!order!of!operations!and!algebraic!properties!to!generate!equivalent!
numerical!and!algebraic!expressions!containing!positive!and!negative!rational!numbers!and!
grouping!symbols;!evaluate!such!expressions.!
10. Represent!real"world!and!mathematical!situations!using!equations!with!variables.!Solve!equations!
symbolically,!using!the!properties!of!equality.!Also!solve!equations!graphically!and!numerically.!
Interpret!solutions!in!the!original!context.!
Grade!8!
11. Understand!the!concept!of!function!in!real"world!and!mathematical!situations,!and!distinguish!
between!linear!and!non"linear!functions.!
12. Recognize!linear!functions!in!real"world!and!mathematical!situations;!represent!linear!functions!
and!other!functions!with!tables,!verbal!descriptions,!symbols!and!graphs;!solve!problems!involving!
these!functions!and!explain!results!in!the!original!context.!
13. Generate!equivalent!numerical!and!algebraic!expressions!and!use!algebraic!properties!to!evaluate!
expressions.!!
14. Represent!real"world!and!mathematical!situations!using!equations!and!inequalities!involving!linear!
expressions.!Solve!equations!and!inequalities!symbolically!and!graphically.!Interpret!solutions!in!
the!original!context.!