Hnl;rD
t.t
v,T
KF_
Replace the Scri pt
"I, W€, You"
with Algebrafied Problems
A man twice as old as his sister weighs 10 pounds more
than his cousin and has 12 more dimes than quarters in
his pocket. He rows 5 miles upstream to get to a candy
store. When he left, a plane left for New York flying
against a 50 MPH headwind. He uses the money to buy
mixed nuts, of which 1O percent are cashews. How soon
after the trains meet does he arrive at the store? What
are the dimensions of the store?
Tom Beatini
E-Mail:
Kelly Cameron
E-Mail:
Union Hill Middle School, Union City, NJ
y
tl24/2OLs
C
L. Establish mathematics goals to focus learning'
2" Implement tasks that promote reasoning and
problern solving.
Use and connect mathematical representations'
cilitatemeaningful mathematical discourse.
3.
4.
5. Pose purpose I questions.
6"
7,
Build procedural fluency from conceptual
understanding.
Support produ ive struEgle in learning
mathennatics.
8. Elicit and use evidence of student thinking.
HC
NATIONAL COU'TCII OF
TEAcHERs oF MATHE'.^ Trcs
NAME
Candy Jar.Problem
Suppose you have a cew candyjarwith the same ratiO of Jolfy
nahlnersi-lR) to_|awbreakers UB) ,s shown in the picture, but it
contains 1S0 JoltY Ranchers-
How many jawbreakers do You have?
Justify your answerNofs
ln
jaurbreakers are
the picii.ne, Jo4ly Ranchers afe rePr.ssented by 5 rectang$es, a*d
shown by 13 circies-
Hg..t|-Thre Cand Jartask. Adapted frorn Smith et al- t2OS5).
-L
t$
nr
t
,1. /.
4-1
5 "-"
){
f n / a,f
-
tn,-.
U \.
! t),1 =
{t'".
l
4)n t f
\ r..',",1 {4A f{Y
t"
I
:l
.'lr1f '
1t
L:
A
E
";5't
f il
jr-t
^
7f,Cir'lr i..{u;r:l
L
Reston, Va.: NCTM' 2014'
National Council of Teachers of Mathematics (NCTM) . Principtes to Actions-
X'un Tees
Version
1
of discount
Fun Tees is offering a30%o discount on all merchandise. Find the amount
on a T-shirt that was originally priced at $16'00'
i o,-r -
rt;
PI
,1
if
/'- t1, ?D
Fun Tees
Version 2
Fun Tees is offering a30Yo discor-rnt on all merchandise'
^.
b.
Find the amount of discount on a T-shirt that was originally priced at $16-00.
c.
l.'
suppose the T-shirt was originatly priced at $17, $18, 19, 20, or $50. Describe
the amount of discount ont-shirts at eachprice'
Write a number sentence that describes how much discount you will receive on
any T-shirt that is offered ata30o/o discount'
'r
L
i't ;5rD
I
;
iC.
I )
)
9.'t \)
<,7
J''"
l.
(', 1 6"00
r) 1100
$Ll. o
Hexagon Pattern Train Task
Trains 1,2,3, and 4 (shown below) are the first 4 trains in the hexagon pattem. The first train in
this pattern consists of one regular hexagon. For each subsequent train, one additional hexagon
is added.
tLl
t0
train
I
1
4lt
n
u, t'
fr
fi
it. I 0 )_
4.
Determine the perimeter of the 25th train without constructing
5.
Write a description that could be used to compute the perimeter of any train in the pattern.
Explain how you know it will always work.
Write an equation that could be used to compute the perimeter of any train in
tl
7.
tte
pattern.
x+
What is the realworld meaning of rate of change?
Extension: How coukl you find the perimeter of a train that consisted of triangles? Squares?
Pentagons? Can you write a general description that could be used to find the perimeter of a
train of any regular polygons?
The Hexagon Train Task has been adapted fiom Vtsual Mathematics
Leaming Center, Salem, Oregon, 19955
nl
^
't
-t
Al
| t(*-l) *.;
n'1\
1rt-t
=
-( J
Fair Booth
Version 1
Science
At
/J
(
2_
Jeremiah and Haley use a piece of rope to mark a square space for their booth at the
science fair. The area of their space is 49 square feet. What is the length of the rope
that Jeremiah and Haley need to use?
Fair Booth
Version 2
Science
Jeremiah and Haley use a piece of rope to mark a rectangular space for their booth at
the science fair. They have a rope that is 28 feet long.
a. What is the area of largest rectangular
space they can enclose? Explain how
you know.
b.
What if the rope was 36 feet long? 48 feet? What would be the area of largest
rectangular space they could enclose? Explain how you know.
c.
What are the dimensions of the largest rectangular space that can be enclosed
with any amount of fence? How did you come to your conclusion?
:)
]-8=
l4:
:54
5:
)
Ll
Lt ito
-r-
5t 12
t sa
= / tir/
iqo
i i;,",rh
.4
Characteristics of mathematical tasks at four levels of Cognitive demand.
Lower-I-evel Demands
Praedrm With Connediou
rization
rula, forrnuh
or definitiqns 1o
memory
'
stude
dwelopirg
. focus
and
n on the use ofgDcerhrrct l-rrr
tlr
plrrpa*1
6f
lsofunderstanding.rlnraiir-rnrricr!crrrx;pts
idec
cannot be sohed using pocedures because a prmedure dms not
in wbjch the task is being completed
exist or b€€N the time
is tm short to use a procedue
. wually
dirctly
represmted in multiple u'ays (e g , visual diagrams,
ections
,,
c
rule
'
re
stated
rc
or
maing
bei ng
I
thal underlie the
amed
or rePnoduced
. require
s
pruedu
Students
sly
e the
edues in mderto successfirlty complete the tasli and &veiop
rstadin€,
ilr! |;rdUrj+ \\r ll h(!ut
Doinp Mathernatics
! rtlll|rarrrlrj
equire cmplex and non-algorilhmic thinking (i
e-,
there is not a
placmenr of the task
requre students to explore md udentand the
mtlematiel
'
have no connection to the
procedue being rced
c
. ae focused on poducing conect
mathematical mderstanding
or m
adog
that urd€die the
mswm nrher
. requue no explamtiom or explmtions thal
desdbing the pnocedure tlat was used
than developing
frcN6
solely on
mtw of
concepts, prtresss, or relationships
demd self-moruluing
or self-regulation of one's oun cognitive
pnEesses
require studenls to accss relevanl knowledge md experiences and
make apprcpiate xe of them in working through the trek
z
t
^o
task
solutions
'require coroidemble tive effort and may involve sone level
of auiety for the student dre to the upedictzble mtm of the
solution Focess Rquired
p. 19
Stein, Smith, Henningsen, & Silver,2000, p
l6