Contents of Grade Level Planning Supports
Page#.
Trapezoid Investigation
2-D or 3-D?
.. So, what is a trapezoid anyway?
• Gulliver Dines with the Mathematicians (Impact Math)
55
How to Trap a Zoid (GSP investigation)
61
Reducing Taxes
• The Mathematicians Transform Rectang.les into Trapezoids
(Impact Math)
64-
Paying Taxes
.. The King moves from Angles to Area (Impact Math)
68
Applying Trapezoid Knowledge
.. Is it Mathematics or Magic? (Impact Math)
73
Performance Task
75
16.1: Area of a Trapezoid
Name:
Date:
Your company has been hired to seal paved highwaysL Sealant is applied in trapezoidal
sections to ensure bonding, As there are curves and intersections, the trapezoids change
and shape for each area. Engineers need to determine the amount of sealant required to cover
any trapezoidal area
Trapezoids are four~sided polygons with two parallel sides, Some examples are provided:
~~
~ I'<~S
r;;;;;:::
:uc
~
§
r:7l
l2!J
la.sl<
Determine a. rule the engineers could use to calculate the area of any trapezoid.
Suggested methods include:
•
Use pattern blocks to construct various trapezoids that are then sketched on dot paper.
•
Draw several trapezoids on dot paper and determine their areas.
•
•
a variety of trapezoids and take useful measurements for calculating
area,
out the trapezoids and cut them further into basic shapes, like squares, rectangles, and
triangles.
Record any numerical data that may help you identify patterns in an organized fashion
the area for any trapezoid, Express your rule as clearly as possible, using
Describe how to
words, pictures, and symbols.
Your process and communication are important for assessment purposes,
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Grade 7
Areas of
for Ontano 2003
(modified by Hcrlsm
; S
Page
I
16.. 2 Assessment Tool: Area of a Trapezoid
Nos: Since this performance task is to be
for inquiry! it is important that students not
have been introduced to finding the area of the trapezoid previously, Students should not be
expected to develop the standard form for the formula for the required area, Rather, any form,
e,g" verba.l, symbolic, or diagrammatic, for the area should be given full
DemoMtratIDl
UaderstaDdiDg
(Understanding
TIPS: Sec!kln
,~Grade
{Areas of
53
Lesson Outline: Trapezoid Investigation
Grade 7
.,IGPICTUBj;
Students
• investigate trapezO!l;Is with without dVl:larnic gelJlm~try sottwlue:
area
•
•
•
measurement
•
•
•
Renel;t on VIH'jOtIS t)lrOp~erWes (J'r trllpezoids,
l:csltnnate areas nf ,Iritino'ipl:
•
K~,,,'i~'N, C~:)I1so!ljjat,;:,
and calc:ulat.e perilnelters and
'3
zero or t.wo
•
TIPS
•
1'\I"1!1)III",
•
tralJCzoid area calculation as a
Queen's Prin1er for On1ario, 2004
t.o calculate
(modified by HCIDSl31
any
Day 1: 2-D or 3-D?
Gradel
g-9dldI9n
MateriaI$
paper
n(ltes
" Distinguish between 2~D
"
on various pr(lpertl(~S oftnape:zoids,
• ESlcimate areas tria'nglt~s and qUBtdfilat€~f'dls,
"• getllnetrk models
1,1, I
I
•
n~lInj~f~!Nandareas~fltri$'lnoj~~
Asussment
Opportunltle.
'mtWul ~ B!!I!W
Students c()mplete Part I
, Circulate and tissist shliJents
and 3-D
the diflierclnce betwe€~n2·D
IDolt!lmYR -+ "item'on
Take up Part of BLM I, i with the
raised,
IDol, Grow -+ Activatt Prior Kn2Wl!*1e
Students select 1-2 shapes from the tist in Question 2
1
liOn
notes.
make a.
of the
chosen and write mle Of
two
that are not included in its def1nitiol1,
the
notes 00 aKWL classmmn
Read
the students'
terms in
and add to the
•
Actionl
PIlI'! ~ InvtltIAfI,Ion
In pairs, students complete Parts
I':!alrs -+ Invutlgation
lsla/lt
and cmnpletc qu(~sH,(ms
and area nf2-i) shapes,),
Curmalum E:lpeetatktftslObnrvattonlRatmg Scale:
effective
Students reite! lhnn
BtJvl .2
to
011
10
Ask pairs to explain how they calculated the areas (,ftlle various
h~
~o
half
Sttldents demonsfnue how
pos:sible answers and (lsk students Wh",'1I1">Y
~, ."""".".~ these
li.)rrnuIIIS,
f:lICO\Jra~tt'
all
more than one
Honw Activity or Further Ca.-room Conaolidation
Write in your Math journals <tbout thernajor di fference between trapei~oi<js
aud the group of
that contains squares,
rhombuses
nfthe
Queen's Prinler for ()ntarlo, 2004
tmCldifi~ad
On/Mo CUJ1'icl'Ilwn
Mil/hematlcs,
Grsdel$ 1·8defines
trapeezold as a
qtlsldriliatel'a! with
Use the rating scale
on BlM 1
allows
the teacher see
Where the students
are in their
Whole CIBa -+ Oiacu..ion
Grade 7 Trapezoids
add
Ihewora
The
ofBLM Ll
"Is a
a 2-D
resiCl0l1id irlch:ldirlg a brief
is an aptmJ'\lm,ate isnsce,les
"[)" connector
the monitor
TIPS Section 3
Select a attldent to
by HG()1SBI
1" 1: So What is a Trapezoid Anyway?
Name:
Date:
Part 1
Think about two~dimensiona! (2~D) shapes and
such as a triangle, lies on a
surface while a
above or below the surface,
three~dimensional (3~D)
1, Write the following geometric objects in
correct column of the table:
3~D
figures, A 2~D shape,
figure, such as a rectangUlar prism, projects
rhombus, right triangle, cylinder, parallelogram, triangUlar prism, square, _ , polygon,
rectangle, sphere, cone,
quadrilateral, pyramid, scalene triangle
ThM:cifm.~rF--",
,,,,,,,,,,,,,--..,...,,,,,,,.,-,,,
~,,--_
....
CC111~u
rhOMbvJ
+ri CU\fj
r1 6 ~+-
+,-i cv;:j I
'Pd.-t--e:LlleA oS ~
Lo_r l'("-,
S v'Vl
I\..Q.
5eu~
ph,
po lA.j<3 00
'P'1~'-d
~le.
ci
LA.,
re-U::.
~ IA....c;...C;,u--; LeA -h:.r-a.-\
..>LJ-',,- ..
Triangle
.-_J'-'-'
~'~ lc:..
RectangUlar
Prism (3~D)
(2~D)
2. Draw a line from each
shape name to its definition. Some definitions could represent
more than one shape so be sure to select the most aptifopriate definition each case,
Polygon
A quadrilateral with both pairs
opposite sides
parallel.
Triangle ......._..-~
"....----A thfee~sided polygon.
A rectangle with all four sides equaL
Parallelogram
A 2~D closed shape whose sides are straight line
segments.
Rectangle
A quadrilateral with all four sides equal.
Rhombus
A four-sided polygon.
Square
A quadrilateral with four right angles and both
pairs of apposite sides equaL
TIPS Section
-
Queen's Printer for Ontario, 2004
{modified by HeClSElI
1.1: So What is a Trapezoid Anyway? (continued)
Part!
The definition that most North American mathematicians use for a trapezoid is: a four-sided
find out about
shape with exactly one pair ofopposite sides parallel. Jake used the
trapezoids, He found several conflicting definitions, They are:
a quadrilateral with two
parallel.
b) a four$sided polygon which has one pair of opposite sides parallel.
a four~sided
shape having one pair of opposite sides parallel, the other pair not
d) a four-sided 2-D closed shape with at least one pair of opposite sides parallel,
answers each question below,
Write the letter representing the definition that
1, Which definition
same as the one that is
by North Americans?
Which def,inition is ambigu<jus, that
it could include quadrilaterals with one or both pairs of
opposite sides parallel? --'lWhich definition includes other quadrilaterals
as a parallelogram or a square? II
e
Note: In the following questions, the definition of trapezoid will be the North American version
given at
beginning of part 2 above,
Part 3
An
trapezoid is one whose non-parallel
are equal
1 Could an
trapezoid be a parallelogram? Explain,
No bec(U).5t! 0.. p:tfodlelo(jram hctS ~ poJfo-l\el 5ides
\<50 fSC C les -tYDfe 7O"f d \t\Gts
Can the parallel sides
0 V\
l ~ .i pcvra(le I Ide.
an iso:sceiles tralDezoid
the parallel sides of any trapezoid
equal? Explain,
a trapezoid ever have
i)
no right angles?
NO
@
Ii) only one right angle?
exactly two right angles?
NO
exactly three right angles?
v) exactly four right angles?
Queen's Printer for Ontario,
Clnd
CLYI
1.. 2: Gulliver Dines with the Mathematicians
(Impact "lIth - Mell$UllImtlnt)
"~)n~~t~~<;; l~suTses of t~Iieb~~~les
int9-.an equJlalt::ral triangle). a pled'e -"
servants cut OUT
cones, cylinden;,
ligures... Their
'1.".,.40'
QafeHt!~mtf'tt:
animal.
Gulliver~ Travels is a popular tale of a traveller named Gulliver who sailed the oceans to
strange and distant lands, Most people know of his
to Lilliput, the land of the little people,
Some know of his visit to Brobdingnag, island of the giants,. But few have read the chapter about
visit are
Gulliver's visit to Laputa, the land of the mathematicians, Some small excerpts from
presented here in a slightly modified form, modernize the old English in which this manuscript
was written almost three
ago!
Activity 1 - Student Page
1. In his description of
dinner, Gulliver confused some 2-dimensional shapes with
3-dimensional figures. Make a list of the 2-dimensional shapes he named and another list of
the 3~dimensional figures. Then rewrite Gulliver's first paragraph using the appropriate
terms, 2.D 'Shap~: e~tA,lalevo.l :r1cu"lgle. J rhDMbus) p~ro.Uelo"ra.rvlS
3l).ft~ures{ Genes, GyLnders
Write a sentence and draw a sketch to explain the meaning of each term,
parallelogram - :t.\AOor i lateral wifh 0rfO~!-r.e s.ld,s pel.f~lte I ,
b) trapezoid ~ '\.,uc.ulrl' lcd-u&t ( wHY! eXIX(.Ai'{""~ f>OII"" of fCtredlel 51 des
equilateral triangle - 0.. friCA-1'\5 l e.. wi~ 0.1\ 'wles o-f 9,(..(.D.\ I.e Yl :lt-t1
rhombus - p£llredlel0'3I"if!ltl" WI t-h 01. tl S·td.es o·f «itA.&\! l.er'\5t-h
-h
\ +:
." J
rectangUlar prism _ po I "1Vtutr"OV\ witi-\ +lAlO pll\.I"Alle / a~ (..Ot\8vuerrl roee-, o.t\Du tty AUSJOII'lt"
f) triangular prism b~ ..{uv.- .... .+o.ces -t\ttov\- Me rIH""l!e(otf"Ul'tS
JO;Vlt>J b.j three ~CI!'S'
3
Name the
2-dimen~oh~~~~*~i~1t~~~e~';~~~":IOW
Count squares to
estimate the perimeter and area of each. Record your estimates.
A', ~h·C1.pe701 d
C', PAraUelojralYl
E..:
TIPS Section :1 ~ Grade
Trapez.olds
Pt{;V'C{
1/ el OJ V"Ct (Y\
Queen'S Printer for Ontario, 2004
(moclified by HeDSB)
1.. 2: Gulliver Dines with the Mathematicians (continued)
4, Write as many of these area formulas as you know,
a) The area of a rectangle given Its length land width w; A':: ,t~w
.Pr I
b) The area of a triangle given Its height h and the length b Its base, A ~ ~ J(
'2
c) The area of a parallelogram given the length lof one
and the perpendicular distance
dfrom it to the other parallel side, A ~ ,l. \( J
Use the formulas you know to
your estimates in
FV'i:>rt"icA
3,
Draw each of
2-dimensional shapes on centimetre paper,
o·f area 20
and perimeter 18 cm,
a
b) a parallelogram of area 24
and perimeter
cm,
a quadrilateral of area
and perimeter 20 cm,
TIPS Section
Queen's Printer for Ontario, 2004
(modified
HeDSB)
1.3: Rating Scale Estimating and Calculating Perimeter and Area
Note: Impact Math - Measurementp, 26 contains estimation techniques and values and the
in exercise 3,
calculated values using formulas for perimeter and area of the 2-D
BlM 1,2
Criteria
Student
in
- knows the fbrmulas Il:lf the area of two
and fecumg!e~s,
calculates
areas llc\:ufllteliy
- uses correct rneasurement units,
able to draw rnost of the
fire reaSOflllble
in
are
in
are
in
rnrnullas Ihr the area
calculatc's most
areas ac(:ur:atelv
- does not use correct measurement
to draw some
reqlle~;ted in
extrtl support}
exllibits Htlle
or
unable to name most of the
in
I.;annot estimate
and area of the
m
does not km)\'! any
area fbmmlas,
the fbnmllas,
cannot calculate the areas
does not use correct measurement
in
unable to dm,;'i any
OUCi'lflS Prlnler for Ontario, 2004
{modified by HGDS!3\
Day 2: How to Trap at Zoid with The Geomettrll Skekhpsd4-
Orade 1
Materials
PmwiPtIon
'" Explore and/or review
for
'"
.BLM
Asse.ment
Small imUR ~ Bra.n8torm
Ask: "What are the similarities
between
and lJeI1C1HlI1O~lJall}er work?" :-)llHlents
eX1JhJl,e g,eollletlry
!!J»1e a•• ~ IbIdmI
with the
Groups share their brtlinstomlil1g
entire class,
Pa.... -7 Guided Exploration
Conduct a teacher-led walkthrough of vl,rious
The (/,'\OI1l'C/{'1'
2,
turns with one student l()cusi,ng on
program,
the Instructions
Students save
use on
~.mDll
StdlWOblerv.tiorllADeedotal:
Whole Class ,7 Sharing
Lead a discusslml blL,Sed on fhe students'
•
l"1<ir,,"ri,~rH'f'
with
Howdid
• Did you have uny
or
with the nro,urlJlm'J
would you like to Jearn more abml!?
•
do you think program like this could be
ihr'J
2,1
• liow
you answer QUlesl:imls
to construct
• How could you use
pfu'all,elo,gram that would
pal'aUtelolsrlUrn when its
1m; dnlgged?
Home AcUviD: Of further Classroom Coneollutlon
• Answer the following questions in your math journals:
me to understand ge(lll1(~try
• How d{)es this program
• What would llik.e to
• How could
TiPS Section
Trapezoids
program be
Quoon's Printer
HCOSBl
2..1: Introduction to The Geometer's Sketchpad'
Using Trapezoids
Name:
Date:
Getting Started
1, Launch
Sketchpad
Click the mouse anywhere to close the introductory window,
3, Maximise both of the
windows.
Notice the six tools at the left of the working area. The second one down is the Point Toot
Click on it, then click in four different places in the work area to make four points.
The fourth tool down is the Segment Toot Click on it then connect the four points with
QAt'1mAnlrQ to
a quadrilateral.
The first tool is the Selection Arrow Tool.
on it then drag the points and segments to
move them around. Try to make your quadrilateral look like a trapezoid.
Follow the directions below to construct a new trapezoid. Once created, the parallel sides of the
trapezoid will remain parallel regardless of how you drag the points or segments.
Constructing a Real Trapuold
Select New Sketch
the
menu.
8. Construct two points and the s.egment between them.
Construct a third point not on the segment.
1
Using the Selection Arrow Tool, select the segment and the third point by clicking on them,
They are highlighted in pink. The original two points should not
11. From the Construct menu, select Parallel Line, You now have a
automatica.lly selected,
1
From the Construct menu, select Point on Parallel Line.
which is forced to always stay on the parallel line.
constructed and
a highlighted point
13. Click the background to deselect the new point
14. Select only the neWly-constructed parallel line and select Hide Parallel Line from the
Display menu.
15 Use the Selection Arrow Tool
the hidden line,
drag the new point around. Notice that you can't drag it off
16 Construct three more segments to finish the trapezoid.
1
Use the Selection Arrow Tool to drag the vertices (points) and
of the trapezoid.
Note that however you drag each point or segment, the two parallel lines always stay
parallel.
18, Drag points andlor sectmE!!nts to make your
look
an
trapezoid
b) a parallelogram
d) a rectangle joined to a right triangle
a rectangle
OU<!len'$ Printer
2"1: Introduction to Thtl Gtlomtlttlr's Sktltchplld'
Using Trapezoids (continued)
Measuring Your Trapezoid
19, Use the trapezoid you created earlier in this investigation,
Click the background to de~select everything,
Using the Selection Arrow Tool, select the four points of your trapezoid in a clockwise or
counter~clockwisedirection,
22, From the Construct menu, select Quadrilateral Interior, Notice
trapezoid
coloured and shelded,
From the M• •ure
left corner the working area,
24. From the Edit menu,
and all Precision
the
Perimeter, Notice that the perimeter
Preferences, On
OK,
shown in the upper
Units tab, set the Distance Units
em
tenths,
Drag the points of the trapezoid to adjust its perimeter to:
a) 25.0 em.
b) 40.0 em,
26. Click the background to de-select everything. Click inside
trapezoid to select it
From the M• •ure menu, select Area,
28, Drag the points
a)
em:?
the
to adjust its area to:
b) 40,0
Can your create a trapezoid with a:
perimeter of 25,0 cm and an area 40,0
perimeter 40,0 em and an area of
When you drag one of the
points that you originally created, another point alwavJt;:
gets dragged along with it Explain why this happens,
When you drag
other points,
TIPS Seclfon
" Gmde 7 ' Trapezo!ds
it moves by
Queen's Printer
Expllain why it
Ontario, 2004
diffierentlv than the
(modified by HCClS8)
Day 3: Reducing Tax..
Orade 7
Material8
RfHrlRtfon
• tJnderstand that a
can
no! one,
.. em
paper
.. st:issors
-HLM :U,:3,2
fornlula fbI' the area of a trll!)ez,oid c011tm:nil1lg two
AS8Msment
Opportunities
Whole CI.- --t Guided DIscussion
Conduct a brief discussion about
tax,
of tl:1XeS, e,g" properlty
Whole elm --t 8ha" Readlna
Consider the "Special Tax" (BL,M 3, I)
students' thitlkjrlg
pnl'binrg 'lIles/ions, such as:
• ttnw can you recogllise
the ma'thelmltllcia;ns reshal)e
bethre and after the tax?
Int have?
mathe:ll1alticilU1S want to
lots un(~hal1gt~d?
insl:ifif,d in
the
of
•
Actionl
Pall1 --t Problem lOWing
In pairs, students complete Questions I and 2 in their notebooks, Di;drihl.llle
grid
to each
any rellJtirmship
fonnd
of a
the U1X and the sum of the
of
the
tax,
them to
how to use
call~ulate the art"a
two
Student!) CIYlllplete Quest!cms 3 and 4 individually, Student,> should discover
that a fine
drawn
of the
A
and B divides it into two
A and
B,
who
that
their
rectangle in
The
where the fold
intersects
A and B is the
which any line
sides CHn be drawn to
the
desired result
Currieulllm ElpedatiouslPerfOmllUlee TasklSeoriDg GWde:
demonstration
Con$o••dam
Debrief
Whole CI-.--t DlscUHion of findIn.
Facilitate student discussions oftheir flndings fhr
and 4,
erllphalsizing that
that there are many ways 1.0 transfimn a
rectangle iruo a
of the same are,L Point outlhat
can have
either zero or two
Home Aetivi1X or Further Cf_room Con8oUdation
In your math journal explain how to l1nd the area of any right-angled
Include an
a fbrmula fi)r the area of a
orils
the dISl:an(:e
between them.
lTape/~oi(:L
Trapezoids
Queen's Printer for
For Msl$tance with
the Impllerru~ntatlon
of this
34·31 in
Indtvldlll--t Guided ~
•
BlM 3.1 and 32 are
from ,_"""" A4"'"
MfJ&stJlCJment p
33
Ontario,
2004
HeDSEl)
MeatitJlCJmfJnl
oofltalrll!1g solutions,
To find the area ota
trapezoid containing
two
3.1: The Mathematicians Transform Rectangles
Into Trapezoids
(Impact Math - MfNI$Ul'flment, Actlvlty 2)
Gulliver observed, with some contempt, that the mathematicians seemed to avoid practical
matters. They
their homes without right angles and located their houses on odd~shaped
lots, Gulliver wa.s apparently unaware of the reasons why
mathematicians constructed their
how the king, in
attempt to
buildings (and their lots) in unsymmetrical shapes. Legend
raise more revenue from his people, levied a special tax on any
that contained more than two
right angles. Two mathematicians, Alpha and Beta, with adjoining rectangular lots, reshaped
their lots as shown, avoid this special tax.
Gulliver proclaimed:
never enjoying a minute's peace of
"These mathematicians are under
mind, for they are always working on some problem that is no
or use to the
are very ill built, the walls bevil, without one right angle in any
of us. Their
apartment; and this defect ariseth from the contempt they
for
geometry.
They despise it as vulgar and impure. ,. Although they can use mathematical tools like
ruler, compasses, pencil, and paper,
are clumsy and awkward in the common
actions and behaviours of life.
By
lots as shown on page 32,
each rectangular lot into a tra),ez'Old.
and
changed
1, a) The diagram below shows two trapezoids, Write a
define a trapezoid.
cruecK your definition with a dictionary,
b) How many right angles has each trapezoid shown
Do all trapezoids have the same number of right angles? t:::xplam
Did Alpha and
have pay the special tax on their new
EXlplain
Q)+rO{)ezo\J: a. <tuadtl\akra\
W\.t\\ ~ G·H V\ ()~ e PC1\''( 0 f
"(}.l(tA \le l si d, s.
b)
Y\O
tlj\\t CH181es
~ t\3V\t
Cl NO
Or-
o.V\5 \6
\:>euUSe 11'Ctft Wtds
More. fW.!t1
J ,. . .''--+f~-------------~._------(moclified by HCDSBj
CD.l'lrtO{ Mil'"
65
GRADE?
AN$WER KEY FOR ACTIVITY
o
a) A trapezoid is a quadrilateral with exactly one
pair of parallel sides. (Sometimes a trapezoid is
defined as a quadrilateral with at least one pair
of sides parallel. This includes parallelograms
as special trapezoids. However, the glossary on
page 84 of The Ontario Curriculum, Grades
1-8 uses the more restrictive form given above.)
b) As seen in the diagrams, a trapezoid may have
no right angles or two right angles. It cannot have
exactly one right angle, for otherwise the other
parallel side would also have a right angle. Also,
it cannot have exactly three right angles because
the remaining angle would have to have measure
90° and then it would have four right angles.
c) Alpha and Beta would not have to pay a tax on
their new lots because the special tax was levied
on the lots with more than two right angles.
Trapezoids cannot have more than two right
angles.
2
@} a) The diagram below shows two rectangles A and
B drawn drawn to scale on centimetre paper.
b) To divide the large rectangle above into
trapezoids C and D so that the areas of A and C
are equal and the areas ofB and D are equal, we
draw a line through the midpoint M of the
boundary between A and B. (We can locate the
midpoint by folding the rectangle perpendicular
to the boundary.) Then we draw any line segment
through M, and it divides the rectangle into
trapezoids C and D.
@ a) The completed tables are shown here.
Since any line through M produces a different
pair of trapezoids, there are an infinite number
ofways ofdividing the rectangle into trapezoids
so that the areas are preserved.
~ Ask students whether the areas remain
~
unchanged if the line segment through
M is allowed to move past the corners as in the
diagram below.
b) The areas are shown in the tables above.
c) No. The areas of the lots remained unchanged.
d) The sum of the lengths of the parallel sides of
Alpha's trapezoidal lot was double the length
of Alpha's rectangular lot. This was also true
for Beta's lots.
e) To find the area of a trapezoid, find the average
length of its two parallel sides and multiply by
the distance between them.
34
You can verify that rotating past the corners does
not preserve areas unless the two rectangles are
of equal area.
3.1: The Mathematicians Transform Rectangles
Into Trapezoids (continued)
a) Measure the length and width in millimetres of Alpha's and Beta's lots before the special
tax was imposed, Record the table on the left.
b) Trace and cut out both lots as they were after the
ta)c Place your cut outs on
centimetre
to determine the area of each
and the lengths of the parallel
Record in the table on the right
Did Al~g and Beta change the areas of the lots when they reshaped them? Explain,
d) Compare the length of Alpha's rectangular lot to the sum of the lengths of the parallel
1ht :~~~f~!~~~~r~~~~i~~lll~rs~;r~fM~~sB~~~~~~(901s~t~~.:L~';l(~~.
, J reetQr13ulM
e) Ex.plain how t~ calculate the area of a trapezoid containing a right angle, given the
. I
}\~Jg~:g~~~~~a~d;fft~d~~t~~~~~r;rt~na~3~~IJrI'~IV'~~ee~r~ttl~f ~t~~~:,{, .
3, a) Draw two rectangles of length 9 em and width 6 cm on centimetre papeL
Divide one of the rectangles into two
and 4 em x 6 em,
A and B with dimensions 5 cm )( 6 cm
what you learned in Exercise
divide the other rectangle into trapezoids C and 0
so the areas of A and C are the same and the areas of Band D are the same. Explain
how
did this. How many ways do you think this can be done?
--to -f\Ni
*" -\rapaolck:
-'/.fOld. 1i1e \,(( tCAV\S Ie
(~~ocO ':>0
t ~ot,( a;c+
(}... hY\ ~ (cv tctSe)~erl)et\cl i 'lY-1 t\ ~
'*'0.
*t
-i'b
b?oul'\d.av- !ir\e Pe.h.tc:~.
CAM ~. nraw \ In e. th(~V\
I.J.··
ls MiafOlr1
"r\'\ 4.
a f Draw a 1
em x
em rectangle on a sheet of paper. Divide your rectangle
two
other rectangles X and Y and record their areas, Cut out your rectangle and divide it into
two trapezoids so that one has the same area as X and the other the same area as Y.
Measure
dimensions of each trapezoid and ca.lculate its area as in 2b. Record the
areas of the trapezoids and verify that they are equa.! to the areas of X and Y.
TIPS Section
Grade 7 Tfapezoids
Queen's Printer IOf Ontiilrio, 2004
(modified
HCDSB)
Ir·(~x5
r
t;
~
3.. 2: Scoring Guide
(Impact Math -- MlINIsursment p. 35)
Scoring Guide for Activity 2
Selection of an
Appr()priate
Strategy for
Constructing li
'frapezoid 'with
Tbe Same Ai"eli as
a Given Rectangle
(t':Xt';~'ClSt';S I)
8)
CONCEPTS
lJrule,'shmds 110"'"
'1'0 Uetermhu.' the
ArC~1
tJ Right
'rrapcloid
~'xerci"c
.!
Queen's Printer for
2004
(moc!ifled by
Grad. 7
Day 4: Paying Tax••
D••crlption
..
Ii
Material•
.. BLM 4. 4.2.1.
formula to calculate the area
4,2J
A••••ment
Opportunlti.
Wb2t! CIaaa ~ .haM Reading
elM 4,1 and 422
are from Impact
Mat!lMeasurement
Students flJllow along as the !ea(~her
the
and poern on BLM 4.1
f)iscuss the
to
I,
the followinil que~sti()Hs:
the
"meanT'
aPI)!"<1iser use the "mean
sideT'
bnse nnd
Review measures of
central tendertcy'
mean. median.
I mode; eS!1;('!cially
"mean,"
•
Actionl
Oplnl1
Plirs.~ GQI. Exgloglol
Students use their trapezoid file for rile (i(;'omeler
stu<lent:s, in
fiS
BLM 4.2.
~2'1
e,im
~ EmIoD\l9n
Pairs work through Questions 2, 3 and 4.
Currieulam ExpeetatioulQuNtioa $lad AuswerlRaug Seale: Students
hand in completed
the end (Jfthe
Currieulam Expeetatioas/Questioa aDd AuswerlRatiDg Seale:
C()lliplet(~d
1 ill based in
oort'lpulier lab
from
work.
421.
~
:2 can
take place in the
classroom
BlM 42.2
Reference
flies
!
I
Who.. Clm ~ Reflection
Facilitate discussion as students
on the
activities. Students share
fbmmlas. Stress
and
common f(Jmmla. Reach <I
Ct)nsensus that
formula
the arel'! of a
could
the average
of the two
sides times the distance between
Unclerstal1l1ing of
number sense.
on::ler of
can be
to confirm
equivalence of
o~miltjofls,
I
formulas that appear
to be different
Home Activity or Further <!Imroom COMolidation
Explain to someone one or two strategics f()r remembering the tOfllnula
area of a
Record any
or discussion items
your convcrsiltion.
Trapezoids
Queer)'$. Printer for
2004
(modffl€id by HenSEl)
6&
4.1: The King Moves from Angles to Area
(ImpactMlllth-lIIt1I11I1Uremtlnt, Activity J)
,. the king levied a special tax on lots with more than two right angles. In response, the
mathematicians reshaped their rectangular lots into trapezoids of the same area. In this way
they preserved the size of each lot and escaped the new tax. The king was not amused, and
sent his
appraiser to announce new
measures,
How did the king
!Illh,::u'l~ of the lot?
the speCil?ll
provision so that
What does the tax appraiser mean by
would not depend on the
side?" by "measurement wide?"
Describe in your own words how the tax appraiser
area of a trapezoid.
Write as a formula the tax appraiser's rule for calCUlating the area of a trapezoid that
parallel sides of length a and b if the
between these
is
think this formula works for a trapezoid that has no right IIIIn(1IA~~?
answer.
o
a) The king based the tax assessment on the area of
the lot instead of its shape, so only the size of
the lot would matter.
b) The mean parallel side is the mean of the lengths
of the two parallel sides of a trapezoid.
Measurement wide means the distance between
parallel sides.
c) The tax appraiser adds the lengths of the two
parallel sides of a trapezoid and divides by 2 to
obtain the mean length of the parallel sides. Then
he multiplies by the distance between the parar 1
sides.
d)
Area = ( a; b) x d
HeeSB)
4.. 2.. 1: Developing a Formula for the Area of Trapezoids
Using The Geometer's Sketchpad 4'J
Name:
Date:
What Do Two Trapezoids make?
Sk~9tcl7Dad jP,
1, Launch
2, Open the
containing the trapezoid you created in Day 2 of this unit
3, Select any
for that
differently.
of the
From the Display menu, choose Color and pick a colour
De-select the side, Colour each of the other three sides of the trapezoid
4. Select one
the non-parallel
the trapezoid. From the Construct
choose
Midpoint,
5. With this midpoint
choose Mark Center from the Transform menu (or simply
double-click on
midpoint),
6. Use Select All from the Edit menu. Choose Rotate from the Transform menu. The angle to
the trapezoid is 1800
7, You
now constructed an
congruent copy ofthe trapezoid, By matching colours,
to which position each of the original segments was rotated.
What type is the resulting shape?
your answer by dragging various points
same or changes to a different type,
noting if the type of shape remains the
Select all of
(comer points) of the
trapeZOid, From
Measure menu to find its area,
choose Quadritateraltnterior, Use
10,
step 9
find the area of the
Construct menu,
figure,
11 What is the relationship between these two areas? Why does this make <;':;l:>l"'l<;t;:.')
12.
the two parallel
Write a formula for the area the ~~:'-::;;';'=",",
terms of h, b1 and
where h is the distance between the two parallel
13. Using information from 11 and 12 above,
write a formula for the area of the original
trapezoid, in
h, bt and b".
Grade'7 Trapezoids
Queen's Printer for Ontario. 2004
(modified by HC()SBj
70
4.2.2: The King Moves from Angles to Area (continued)
(lmpscf Mlltb .... MHSu.nJf1H1nf, ActivIty 3)
2. a) Is the tax appraiser's rule for calculating
the area of a trapezoid the same as the
formula. you discovered in Activity
Explain your answer.
b) Use the tax appraiser's rule to calculate
the areas of the trapezoids drawn on
this centimetre grid.
3.
Draw a line
to divide trapezoid A in
2
a right triangle
a
area trape.:zoid
rectangle. Calculate the areas of the rectangle and triangle to find
A Compare with
answer in
b) Divide trapezoid B in Exercise 2 into two triangles. Then use the formula for the area of a
triangle to calculate the area of trapezoid 13, Compare with your answer in Exercise
4. a) Draw a trapezoid
the one on the right on
paper and
count squares to determine its area. Draw another trapezoid congruent
to it Cut out both trapezoids and fit them together to form a rectangle.
Record the area of the rectangle and the area of each trapezoid in 461.
A congruent copy the trapezoid below is made and they are fitted
tog.ether to form a rectangle as shown,
Write an expression for the area of
of 8, b, and d
A congruent copy of the trapezoid below is
parallelogram as shown,
and for the area of each trapezoid
and they are fitted together to form a
Challeng.e:
Write an expression
8. b. and d Show
TIPS Section
the area of the parallelogram and for the area
work
- Grade 7 l'rapezoids
Queen's Printer
terms
Ontario. 2004
(modified
the tral:>Elz:old in
,./
GRADE?
AN8WER KEY FOR ACTIYITY
o
3
a) The area of the given trapezoid is 9 cm 2•
b) The area of the rectangle in Oa is 6 cm x 3 cm
or 18 cm 2• Therefore the area of each half is 9
cm2 and this is the same as our answer in Oa.
c)
@ a) Yes, the sum of the lengths of the parallel sides
divided by 2 is the mean parallel side.
b) Area ofA = 2 x 3 = 6 cm 2•
Area ofB = 5/2 x 4 = 10 cm 2•
The length of the rectangle is a+ b.
The width of the rectangle is d.
The area of the rectangle is (a + b) x d.
The trapezoids are congruent and therefore equal
in area. The area of each trapezoid is half the area
of the rectangle, i.e.,
Area
Area of A
= (a+b)xd
2
-.
(a+b)
or Area = -2- x d
= Area of rectangle + area of triangle
= 3 cm2 + 3 cm2
=6cm 2
This is the same as the area calculated in @ b.
b) We draw a diagonal of the trapezoid B to divide it
into two triangles.
Area of B = Area of triangle I + Area of triangle 2
= 1/2(3 x 4) cm2 + 1/2(2 x 4) cm2
= 6 cm2 +4cm2
= 10 cm2•
Alternatively, we could have drawn the other
diagonal and achieved the same result. This area
matches that found in @ b.
42
The base of the parallelogram is a+ b.
The height of the parallelogram is d.
The area of the parallelogram is (a + b) x d.
The trapezoids are congruent and therefore equal
in area. The area of each trapezoid is half the area
of the parallelogram, i.e.,
Area=(a+~)Xd or
Area=(a;b)Xd
4.. 2.2: The King Moves from Angles to Area (continued)
(lmPllel Math -lIIfJII8ulflmtmt}
AorMTY 4' -
o
8rVIJI'Nr
PIltM
the
a)
1m the
I 'UlflUl1llll
unit.
ottmlug.lcs Beta and Gamma?
Whlit do you notice
c) Arc
ntJL
not
d) Arc trap;ezoids
the lirollS in your table to I1nd the tnlal [Ireu of all four lob;,
\\Fha! is
",''''''i>N>
ofthis rectangle
tlltal
your llnSWet'S in 0
and @
and eXllla1H
cl'.lnlimCi.rc paper. cut oul
yoUf tables. Show
have total area of 64 cml
loti( in
did the extra unil
An'ufIl!c Ihellc
rCll$On" to
,.)"....,,,,, and Deltli with the dimensions
In
arrangIng them in an 8 cm 8 ein square.
tllld width em . What
lhe lotl/larClt or the 101:<;"
64 sillw.rc uuits Of 6~ squlIfe
"our ll.l"li,UfltCnt
Indicate willit
orthe lolal tax should l:llt ll:'4sigrlt:u
of the lbur lols
--------------------------------_.
-:5
TIPS Section
.~
Grade
Queen's Printer for OntariO, 2004
4.10.3: Developing a Formula for the Area of Trapezoids
Using The Geometer's Sketchpad(@)4
Name:
Date:
What Do Two Trapezoids Mak.e?
1. Launch The Geometer's Sketchpacfll'4,
2. Open the file containing the trapezoid you created in Day 9 of this unit
3. Select any side of the trapezoid. From the Display menu, choose Color and pick a
colour for that side, De~select the side. Colour each of the other three sides of the
trapezoid differently,
4. Select one of the non~parallel sides of the trapezoid, From the Construct menu,
choose Midpoint.
5, With this midpoint s.elected, choose Mark center from the Transform menu (or
simply double~Uck on the midpoint).
6, Use Select All from the Edit menu, Choose Rotate from the Transform menu, The
angle to rotate the trapezoid is 1800 ,
7. You have now constructed an exact, congruent copy of the trapezoid, By matching
colours, notice to which position each of the original segments was rotated.
8, What type is the reSUlting shape?
Test your answer by dragging various points and noting if the type of shape remains
the same or changes to a different type.
9. Select all of the vertices (comer points) of the orjginal trapezoid. From the Construct
menu, choose Quaddlaterallnterior, Use the MeasYre menu tQ fincjits Eirea.
10. Repeat step 9 to find the area of the entire figure.
11. What is the relationship between these two areas? Why does this make sense?
12. Label the two parallel sides b t and~. Write a formula for the area of the whole
shaRe, in terms of h, b i and b2, where h is the distance between the two parallel
sides,
13. Using information from 11 and 12 above. write a formula for the area of the original
trapeZOid, in terms of h. b 1 and b2•
GRADE?
AN8WER KEY FOR ACTII/ITY
o
a) The completed tables are shown below.
b) The areas of Beta and Gamma are equal.
c) Beta and Gamma are right triangles with legs of
length 8 and 3. Two such right triangles are
congruent because they have two sides and a
contained angle respectively equal. Alternatively,
students may observe that they are congruent
because they can be placed in coincidence.
d) Trapezoids Alpha and Delta are congruent
because either can be cut out and placed in
superposition with the other.
e) The sum of the areas of the four lots as shown in
the table is 12 + 12 + 20 + 20 or 64 square units.
@ a) The rectangle containing the four lots is 13 units
long and 5 units wide, and so has an area of 5 x 13
or 65 square units.
b) The areas of the lots given in the table have a
total of 64 square units, but the rectangle formed
by the lots has an area of 65 square units. Therefore
the tax collector does not know whether to assess
an area of 64 square units or 65 square units. That
is why he was confused.
I 1IA(JII~R NOi~ I
The apparent paradox presented above is a classic visual
trick. When the lots are arranged in an 8 x 8 square, they
fit together without spaces or overlap. However, when
rearranged to form a 13 x 5 rectangle, the lots appear to
fit together but, as students may discover in Exercise.,
there is a small gap along the main diagonal that has a
total area of one square unit.
50
4
C} a) The students will display this:
b) When the students put the pieces together to fonn a
rectangle, there will be a sliver of space of area 1 cm 2
along the main diagonal. The reason for this is that when
the rectangle is partitioned into lots Alpha, Beta,
Gamma, and Delta, these lots are all quadrilaterals. Beta
and Gamma are not triangles because line segments
PQ, QR, and RS have different slopes. To find the areas
of lots Beta and Gamma, we must break each into a
trapezoid plus a triangle.
T
Area of lot Beta
= Area oftriangle QWR + Area of trapezoid TWRS
=3!2cm 2 + 11/2 x2 cm 2•
= 12.5 cm 2
Similarly, the area of lot Gatmna = 12.5 cm 2•
Together, lots Beta and Gamma have a combined area
of 25 cm 2, which is 1 cm more than the triangular lots
Beta and Gamma. The extra 1cm 2 came from reshaping
the triangular lots Beta and Gamma into quadrilaterals
of slightly larger area.
TIIK ApPI'/AI$~R'$
tlM~RICK
The area formulas for the triangle and parallelogram
are special cases of the area fonnula for trapezoids. The
idea that formulas remain unchanged as one shape
changes into another is a powerful concept that pervades
mathematics and motivates the continual quest for
general forms. At this level it's just an interesting
curiosity for kids.
Gradel
•
A.....m.nt
!fb21e CI_ -+ lharing
Student volunteers share their journal entries
the flr€~Vlfm'l
answer somt~
'''''''''" '.11'" "''' the conversation,
Whole CIM! -+
BlM !i 1 ll, from
RtvW
Briefly review
discussed on the
IndJ!klull =t Performanct IMk
Students complete Questions I :2 and
including the
Cit'culatl':' to ensure students
StudertL<; c'DlT!:plcite BLM
to
did
Hom! As;tIvD! or Funn-r CllftfOom Conaolidd2n
Record
in your home environment where trapezoi.ds occur, Answer Ihe
in your math !ot!rn~l.l:
..
are
common?
.. Where do you find
in your home')
tfllitO\\!!fIQ qm~stJons
M
Sometime
Actlonf,
wish to il1c1orp,orate
brain gym l'lctivitiias
such as SlfE!tctllng
and blood·flow
lechrdques..
Whgft CI_ =t 8barlng
• What did students find ditlieult?
.. What was stf<3igihltilr\A1llr<l'!
•
Clln students
upon whnt
BlM 52 from
TIPS De,,'ek1lolng
MathemaHcaI
Prooo$ses
Grade 7
Example of
trapezoids: Wes near
the
of
walls or the area
betwelEinthe
of III kJtchen
or
other furniture with
stUdents
trapezoids in
patterns can
serve as an effective
to the
folk:;wifl{j section
TIPS Section 3
Grade
Trapezoids
Queen's Printer
5.1: Is It Mathematics or Magic?
(/mpllct MlltII- MtllI6UlflJmenf, Activity 4)
learned Activity 3
particlilarly proud
centimletre on
TIPS
Queen's PrInter for Ontario, 2004
(modmed by HeDSB)
5.. 2: Application of Trapezoid Area and Perimeter
Name:
Date:
1. Reaaonlng and Proving
2. Reasoning and Proving
3. Communicating
4. Making Connectiona
2m
Palio Doo,rway
Queen's Printer for Ontario, 2004
(mocUfred
Gradel
5.. 3: Performance Task - Evaluation Rubric
Problem
Solving
awareness
there is a parad{IX
in 5,1
Grade 7
Level 4
Level 2
is unaware that at
one
in 5,1 has increased
in area but
nlhef\\m:e applies
- is aware that at
one ofthe lots
in 5.1
in area
applies some area
cul(~ulaJe
M.aking
Connedions
solid cormecti()lls
to
euoniDg
and Proving
to
the
(~v(·r,!.'lllV
answers are llrrillefj
cmnp1eteltlcy in
eXIHulllllrlg answers
- uses Iml'lted
proper tennirlO!\lgy
when eXt1lainil11g
TIPS
ext)!aining answers
Queen's Printedor
2004
(modified by HCr)Sfn
76
Lesson Outline: 2D and 3D Relationships
Grade 7
•
nets to construct
structures,
•
TIPS Sechon
Grade
3D
Queen's Printer lor Ontario 2003
(modifIed by HCDSSj
4.10.3: Developing a Formula for the Area of Trapezoids
Using The Geometer's SketchpacfID4
Name:
Date:
What Do Two Trapezoids Make?
1. Launch The Geometer's SketchpacP4.
2. Open the file containing the trapezoid you created in Day 9 of this unit.
3. Select any side of the trapezoid. From the Display menu, choose Color and pick a
colour for that side. De-select the side. Colour each of the other three sides of the
trapezoid differently.
4. Select one of the non~parallel sides of the trapezoid. From the Construct menu,
choose Midpoint.
5. With this midpoint selected, choose Mark Center from the Transform menu (or
simply double-ellck on the midpoint).
6. Use Select All from the Edit menu. Choose Rotate from the Transform menu. The
angle to rotate the trapezoid is 1800.
You have now constructed an exact. congruent copy of the trape.zoid. By matching
colours, notice to which position each of the original segments was rotated.
8. What type Is the resulting shape?
Test your answer by dragging various points and noting if the type of shape remains
the same or changes to a different type.
9. Select all of the vertices (comer points) of the original trapezoid. From the Construct
menu, choose QuadrUaterallnterior. Use the Measure menu to find Its area.
Is the relationship between these two areas? Why does this make sense?
12. Label the two parallel sides b, and b2' Write a formula for the area of the whole
sha~, in terms of h, b, and b2" where h is the distance between the two parallel
sides.
13. Using information from 11 and 12 above, write a formula for the area of the original
trapezoid, in terms of h, b1 and b2,.
h