proportional reasoning: focus on sense-making
TRANSCRIPT
Proportional Reasoning:Focus on Sense-Making
Chris Hunter Numeracy Helping Teacher
Surrey Schools
twitter:@ChrisHunter36
email:[email protected]
the goods:reflectionsinthewhy.wordpress.com/bcamt2016
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Big Idea(s)
Compete
ncies Content
Big Idea(s)
Compete
ncies
ratios rates
proportions unit price percent coupons
Content
Big Idea(s)
Compete
ncies
ratios rates
proportions unit price percent coupons
Content
Proportional reasoning helps us make sense of multiplicative
relationships.
Big Idea(s)
Compete
ncies
ratios rates
proportions unit price percent coupons
Content
Proportional reasoning helps us make sense of multiplicative
relationships.
Use multiple strategiesto solve problems
Communicate in a variety of waysto explain and justify ideas
Big Idea
Students will understand that • Thinking about how quantities are related using multiplication is essential for solving a wide variety of
problems • Ratios, rates, and percent make comparisons easy; one term is made the same
Curricular Competencies Content
Students will be able to: • choose correct and efficient strategies • monitor progress to completion of task and make
necessary adjustments along the way • propose and consider or critique alternative
strategies • share mathematical ideas–not just steps!–needed to
solve problems (verbal & written) • present work that is clear and easy to follow • effectively use tables, equations, etc. to support
conclusions or arguments
Students will know that: • two equivalent ratios represent the same
relationship • ratio tables list equivalent ratios in an organized way • a rate represents an infinite number of equivalent
ratios • a unit rate (or price) is an equivalent rate where one
term is “1” • a proportion is an expression of the equivalence of
two ratios • proportion problems can be solved by looking for
scale factors within or between ratios • a percent is a fanatical comparison to 100
Big Idea(s)
Compete
ncies
ratios rates
proportions unit price percent coupons
Content
Proportional reasoning helps us make sense of multiplicative
relationships.
Use multiple strategiesto solve problems
Communicate in a variety of waysto explain and justify ideas
Use l
ogic
and p
atter
ns to
solve
puzz
les an
d play
game
s
Use r
easo
ning a
nd lo
gic to
explo
re, a
nalyz
e, an
d app
ly ma
thema
tical
ideas
Estim
ate re
ason
ably
Demo
nstra
te an
d app
ly me
ntal m
ath st
rateg
ies
Use t
ools
or te
chno
logy t
o exp
lore &
crea
te pa
ttern
s &
relat
ionsh
ips, &
test
conje
cture
s
Mode
l math
emati
cs in
conte
xtuali
zed e
xper
ience
s
Apply
mult
iple s
trateg
ies to
solve
prob
lems i
n both
ab
strac
t and
conte
xtuali
zed s
ituati
ons
Deve
lop, d
emon
strate
, and
apply
math
emati
cal
unde
rstan
ding t
hrou
gh pl
ay, in
quiry
, and
prob
lem
solvi
ng
Visu
alize
to ex
plore
math
emati
cal c
once
pts
Enga
ge in
prob
lem-so
lving
expe
rienc
es th
at ar
e co
nnec
ted to
plac
e, sto
ry, cu
ltura
l pra
ctice
s, an
d pe
rspec
tives
relev
ant to
loca
l Firs
t Peo
ples
comm
unitie
s, the
loca
l com
munit
y, &
other
cultu
res
Use m
athem
atica
l voc
abula
ry &
langu
age t
o con
tribute
to
mathe
matic
al dis
cuss
ions
Expla
in an
d jus
tify m
athem
atica
l idea
s and
decis
ions
Comm
unica
te ma
thema
tical
think
ing in
man
y way
s
Repr
esen
t math
emati
cal id
eas i
n con
crete,
picto
rial, a
nd
symb
olic f
orms
Refle
ct on
math
emati
cal th
inking
Conn
ect m
athem
atica
l con
cepts
to ea
ch ot
her a
nd to
oth
er ar
eas a
nd pe
rsona
l inter
ests
Use m
athem
atica
l arg
umen
ts to
supp
ort p
erso
nal
choic
es
Incor
pora
te Fir
st Pe
oples
wor
ldview
s and
persp
ectiv
es
to ma
ke co
nnec
tions
to m
athem
atica
l con
cepts
A B C D E F G H I J K L M N O P Q R
perfect squares and cubes 1square and cube roots 2
percents less than 1 and greater than 100 (decimal & fractional %) 3numerical proportional reasoning (rates, ratio, proportions, & %) 4
operations with fractions (addition, subtraction, multiplication, division, & order of operations) 5
discrete linear relations (extended to larger numbers, limited to integers) 6
expressions- writing and evaluating using substitution 7two-step equations with integer coefficients, constants, and solutions 8
surface area and volume of regular solids, including triangular and other right prisms and cylinders 9
Pythagorean theorem 10construction, views, and nets of 3D objects 11
central tendency 12theoretical probability with two independent events 13
financial literacy — best buys 14
Big
Idea
s
*Number represents, describes, and compares the quantities of ratios, rates, and percents.
*Computational fluency and flexibility extend to operations with fractions.
*Discrete linear relationships can be represented in many connected ways and used to identify and make generalizations.
*The relationship between surface area and volume of 3D objects can be used to describe, measure, and compare spatial relationships.
*Analyzing data by determining averages is one way to make sense of large data sets and enables us to compare and interpret.
Cont
ent
Grade 8 Math Curricular CompetenciesReasoning and Analyzing Understanding and Solving Communicating and Representing Connecting and Reflecting
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MARS
Rod Ratios
What is the ratio of this pair of Cuisenaire Rods? How do you know? How many pairs can you find with the same ratio?
@robertkaplinsky
Proportional Problem Types:(1) missing-value, and (2) comparison
Split Time
@ddmeyer
Split Time
What’s the first question that comes to your mind? What’s a guess that’s too low? What’s a guess that’s too high? Write down your estimate. What information would be helpful to know here?
Split Time
Split Time
Representation: Ratio Table
metres 400
seconds 75
Representation: Ratio Table
metres 400 40
seconds 75 7.5
Representation: Ratio Table
metres 400 40 80
seconds 75 7.5 15
Representation: Ratio Table
metres 400 40 80 160
seconds 75 7.5 15 30
Representation: Double Number Line
seconds
400
75
0
0
40 80 160
7.5 15 30
metres
Carnival Tickets
What questions do you have?
@robertkaplinsky
What do you notice?
What do you wonder?
“Best use of textbooks ever.”
M
athChallenge
25#
Answer:Good Grape should have the strongest grape taste.
Ratios are fractions that compare two or morequantities. Shoppers use ratios to compare prices;cooks use them to adjust recipes. Architects and
designers use ratios to create scale drawings.
Figure This! If all grape juice concentrates are the same strength, which recipe would you
expect to have the strongest grape taste?
???
?
GR
APE JUICE JUNG
LE
Hint: For each recipe think about how much watershould be used with 1 cup (c.) of concentrate,or how much concentrate should be used with
1 cup of water.
Which tastes
JUICIER
NCTM
Which city is “selfier”?
Anaheim, California Milan, Italy
reflectionsinthewhy.wordpress.com/wncp-virtual-file-cabinetMore Missing-Value & Comparable Comparison Problems
@robertkaplinsky
Curricular Competencies RevisitedBig Idea(s)
Compete
ncies
ratios rates
proportions unit price percent coupons
Content
Proportional reasoning helps us make sense of multiplicative
relationships.
Curricular Competencies RevisitedBig Idea(s)
Compete
ncies
ratios rates
proportions unit price percent coupons
Content
Proportional reasoning helps us make sense of multiplicative
relationships.
?