Lesson 2: Illustrating
Ratio and Proportion
Elicit 1. Express Express the the following following as as ratios: ratios: (a) ½, ½, (b) 2/5, (c) (c) 5/6 5/6 2. In the followin following g ratios, ratios, how an! ties ties is is one "al#e "al#e as large large as the other$ other$ (a) % : & (b) 12 : 6 (c) ' : &
Refect: ow i !o# express fractions as ratios$
Engage *hat is the ratio of the n#ber of white to the n#ber of blac+ arbles of lfie$ *hat abo#t -ac+ie, what is the ratio of the n#ber of white arbles to the n#ber of blac+ arbles$ re the ratios e#al$
Alfe’s marbles
Jack’s marbles marbles
Refect:
*hat o !o# want to +now abo#t ratio an proportion$
Explore oo+ at the two fraction bars at the right. *hat fractions o their shae parts show$ re the two fractions e#al$ ow o we show that the two fractions are e#al$ 1 3
2 =
6
ow o we express the fractions abo"e in ratios$ re the two ratios e#al$ ow o we express that two ratio are e#al$ 1 3
→
1 :3
2
an
6
→
2: 6
*e #se e#al s!bol or 0 also to show that two ratios are e#al. 3he two e#al ratios below show a proportion. 1 : 3 =2 : 6
*hat ratio represents the n#ber of circles an the n#ber of iaons in the first fig#re$ ow abo#t in the secon fig#re$ re the two ratios e#al$ ow o we express ex press two e#al ratios$ 4ne of the an! #ses of proportion is in ba+ing or coo+ing a recipe. 3he table shows ratios of the ao#nt of ingreients in a recipe.
No of Serving 1 2 ' % 5
Eggs
1 2 ' % 5
Cups of flour % & 1119 2 16 2
*hat are the two #antities being copare$ *hat happens to the ingreients as the n#ber of ser"ing increases$ Is the n#ber of eggs proportional to the n#ber of c#ps of flo#r$
Explain elow are the ieas !o# sho#l ha"e learne in this lesson. 7an !o# explain each one of the with one of !o#r classates$ 7hec+ those !o# #nerstoo well an cross o#t those where !o# nee f#rther explanation. 1. proportion is the e#alit! of two ratios. *hen two ratios are e#al, the! are sai to be proportional. 2. If two ratios are proportional, the n#ber of ties one "al#e is as large as another is the sae for the two ratios. 8or exaple, ' : 6 12 : 2%. In the first ratio, 6 is 2 ties as large as ' an in the secon ratio, 2% is also 2 ties as large as 12. '. If the ratios are expresse as fractions, we can #se the e#al () s!bol to inicate the 1
% =
2
&
proportion. 8or exaple, is a stateent of proportion. *e sa! that ½ is proportional to %/&. 9ecall that in the past lesson, we can chec+ that two fractions are 1
% =
2
&
e#al b! getting the crosspro#cts, that is, since 1 x & 2 x %, therefore the fractions are e#al an the stateent is proportional. %. If the ratios are expresse #sing the colon s!bol (:), it is ore appropriate to #se the 1
% =
2
&
o#ble colon (::) instea of the e#al s!bol. 8or exaple, can be also written as 1:2 :: %:&. *e rea this as ;one is to two as to fo#r is to eight<. owe"er, it is still correct to write the proportion as 1:2 %:& which can be rea as ;one is to two e#al %:&. 3hese three representations of a proportion are all correct. 5. *e can represent a proportion with a fig#re. =sing triangles an s#ares, we can raw the fig#re corresponing to the proportion 1:2 :: 2:%, as follows:
Elaborate nswer the following. In !o#r own wors, explain what a proportion is. 1. >i"en the ratio 2:', fin other ratios that for a proportion with it. *rite the proportion in three ifferent wa!s #sing s!bols. Express the proportion in wors. 2. Why is the statement 3: :: !:9 "#$ a proportion% &hange one element 'or one o( the numbers to make the proportion correct% '. Explain the stateent: 3he ao#nt of har"est in a ango far is proportional to the n#ber of ango trees. %. >i"e reallife exaple of a proportion.
Evaluate
1)*
A. ?o the following. 1. Ientif! the ratio/s is/are @43 e#al to ':%$ a. 15:2 b. 6:& c. 12:1& . 1&:2% e. ':% 2. Ahow whether or not the following is a proportion: a. 2:' :: %:6 b. 2:B 6:21 c. 5:C :: 1&:1 . B:5 :: 21:15 '. 8in three ratios that are proportional to 2:5. B. nswer the following. Explain !o#r answers. 1. ow o we test whether two ratios are proportional$ 2. *h! is 2:' not proportional to ':%$ '. If two ratios are proportional to a thir ratio, are the two ratios proportional$ >i"e exaple. %. Dichelle an her sister hriFna sells a"ocaos an santol. In the bas+et of Dichelle, there are ' a"ocaos an B santols. In hriFnas bas+et, there are 12 a"ocaos an 2& santols. ?raw a fig#re to represent the sit#ation. re the n#bers of a"ocao an santol in the bas+ets proportional$ C. A+etch a fig#re that oels the following proportion 1. %:5 &:1 2. 6:& ':% '. 1: 1 2:2 D. 8ollow the irections to a+e a proportional siFe of the pict#re below. 1. 7onstr#ct a 1 b! 1 gri on a piece of paper whose inter"al is thrice as large as that of the gri of the pict#re. 2. 7op! the pict#re of o#r national hero on the larger gri. a. *ho is in the pict#re$ b. Is the pict#re !o# rew proportional in siFe to that of the pict#re in the boo+$ Explain or show proofs. Pause & Refect
re !o# happ! with the res#lt of !o#r e"al#ation$ *h!$ *hat concepts are clear to !o#$ *hat concepts are still not clear to !o#$ *hat is the ost esirable "al#e or attit#e that !o# ha"e e"elope in this lesson$
Exten+ At#! the table on the right which shows the ao#nt of gasoline for e"er! +iloeter istance tra"ele b! a bran new car on brea+in. a. *hat is the ratio of the ao#nt of gasoline to the istance tra"ele$
o#nt of >as () 1 2 ' % 5 6
?istance tra"ele () 1% 2& %2 56 B &%
1)1
b. Is the istance tra"ele proportional to the ao#nt of gas #se$ *h!$ Ill#strate with exaples b! setting sets of proportion.
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