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Learners Material

Module 5

Department of EducationRepublic of the Philippines

7

This instructional material was collaborativelydeveloped and reviewed by educators from public andprivate schools, colleges, and/or universities. We encourageteachers and other education stakeholders to email theirfeedback, comments, and recommendations to theDepartment of Education at action@deped.gov.ph.

We value your feedback and recommendations.

mailto:action@deped.gov.phmailto:action@deped.gov.ph

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Mathematics Grade 7Learners MaterialFirst Edition, 2013ISBN !7"#!71#!!!0#$0#%

&e'()lic *ct "2!3, section 17$states that: No copyright shall subsist in any workof the Government of the Philippines. However, prior approval of the government agency oroffice wherein the work is created shall be necessary for exploitation of such work for profit.uch agency or office may, among other things, impose as a condition the payment ofroyalties.

!orrowed materials "i.e., songs, stories, poems, pictures, photos, brand names,trademarks, etc.# included in this book are owned by their respective copyright holders.$very effort has been exerted to locate and seek permission to use these materials fromtheir respective copyright owners. %he publisher and authors do not represent nor claimownership over them.

Published by the &epartment of $ducationecretary: !r. 'rmin '. (uistro )*+ndersecretary: olanda . -uiano, Ph.&.'ssistant ecretary: $lena /. /ui0, Ph.&.

+rinted in the +hili''ines ) ------------

.e'artment o/ Ed(cation#Instr(ctional Materials o(ncil Secretariat .e'Ed#IMS

1ffice 'ddress: 2nd)loor &orm G, P* *omplex3eralco 'venue, Pasig *ityPhilippines 4566

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4a)le o/ ontents

Statistics

(esson 7;: =ntroduction to tatistics @@@@@@@@@@@@. 277(esson 7A: 1rgani0ing and Presenting &ata @@@@@@@@... 27(esson 7B: 1rgani0ing and Presenting &ata +sing )reCuency

%able and Histogram @@@@@@@@@@@@@... 284(esson 86: 'verages: 3ean, 3edian, and 3ode @@@@@@@ 28(esson 84: 'naly0ing, =nterpreting, and &rawing *onclusions

from Graphics and %abular Presentations @@@@... 28A

"

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Lesson 37 Introd(ction to Statistics

+rere6(isite once'ts 3easurement concepts, length measurements, measurement units

*)o(t the Lesson

%his is an introductory lesson to basic concepts, uses, and importance of tatistics.%he first lesson allows you to experience systematic gathering and organi0ing data. %his

makes use of your knowledge of arranging numbers according to some considerations, like

arranging numbers in descending or ascending order.

)8ectie

=n this lesson, you are expected to:

4. *ollect or gather statistical data and organi0e the data in a freCuency table according

to some systematic considerationD

2. $xplain the basic concepts, uses and importance of tatisticsD and

7. Pose Cuestions and problems that may be answered using tatistics.

Lesson +ro'er

I9 *ctiit

http:EEwww.gnbco.comEbowFlength.php

&o you think students in this class have different arm spans How many in this class have

the same arm spans hat is the most common measure of arm spans %o answer these

Cuestions, you will to do the following:

Instructions:

4. +sing a tape measure or a meter stick, measure your individual arm span. +se the

centimeter "cm# unit of length. /ound off measures to the nearest cm.

2. 1n the board, write your measures individually.

Questions:

4. hat do these numbers represent

#

3easuring the arm span: tretch out botharms and measure the length from the tip ofa middle finger to the tip of the other middlefinger.

http://www.gnbco.com/bow_length.phphttp://www.gnbco.com/bow_length.php

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2. *an we get clear and precise information immediately as we look at these numbers

7. How can we make these numbers meaningful for anyone who does not know about

the description of these numbers

II9 *ctiit

!ased on the previous activity, after listing all your measures on the board, group yourselvesin to groups of to A members. ort out the raw data and present it in a way you think would

be a good presentation. 'fter 46 to 4 minutes, present your output.

III9 :(estions;+oints to +onder

4. =n our daily activities, we encounter a lot of sorting and organi0ing obects, data, or

things like what you ust did. %hese are ust few of the activities of doing tatistics.

a. hat are some of the few activities that you did in the previous activity

b. hat is tatistics

2. Give some examples of activities which you think tatistics is involved.

7. (ist down some problems or Cuestions that can be answered using tatistics.

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Lesson 3" r>ani?in> and +resentin> .ata

+rere6(isite once'ts /atio and proportion, measurement concepts

*)o(t the Lesson

%his lesson allows you to explore different ways of organi0ing and presenting datasuch as using tables, graphs or charts. Presenting data using graphs or charts such as

freCuency histogram, bar graphs, line graphs and pie charts or circle graphs will be studied.

%his will help you reali0e when to use such kind of graph and what information each of these

types can provide.

)8ectie

=n this lesson, you are expected to:

4. 1rgani0e data in a freCuency table

2. +se appropriate graphs to represent data.

Lesson +ro'er

I9 *ctiit

*onsider the following charts and graphs and answer the Cuestions that follow:

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Population of tudents in !"##

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4. hat information can we get from each of the above charts or graphs &iscuss each

chart or graph.

2. &o they present the same information

2. &escribe each of the chartsEgraphs. hat do you think are some uses of each of the

charts or graphs

II9 .isc(ssion

' histo>ramis a graphical representation showing a visual impression of the

distribution of data. ' histogram consists of tabular freCuencies, shown as

adacent rectangles, erected over intervals. %he height of a rectangle is also eCual to the

freCuency.

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http:EEwww.mathsisfun.comEdataEhistograms.html

* 'ie chart is a circle divided into sectors proportional to the freCuencies. =t showshow a part of something relates to the whole. =t is important to define what the whole

represents.

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* )ar >ra'his like a histogram except that its bars are separated. %his uses parallelbars, either hori0ontal or vertical, to represent counts for several categories. 1ne bar isused for each category with the length of the bar representing the count for that onecategory.

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http:EEwww.k42.hi.usEJgkolbeckEwebsiteEgrade2.html

* line >ra'h shows trends in data clearly. %his displays data which are collectedover a period of time to show how the data change at regular intervals.

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http:EEhwachongilp.wikispaces.comE;.IHowItoIinterpretIlineIgraphs

III9 *ctiit

4. 1rgani0e the following data and present using appropriate graph or chart. $xplain whyyou are using such graphEchart in presenting your data.

a. %he data below shows the population Kin thousandsL of a certain city.

ear 4B;

4BA6

4BA

4BB6

4BB

2666

266

2646

Populationin thousand 5 ;A A6 A4 A2 A5 B6 426

b. %he following data indicates the scores of 76 students who took the Cualifyingexamination for mathematics challenge.

7, 56, 8, 5, 8, 26, 5, A6, ;6, 78, 72

B6, B, ;6, 28, 44, ;6, 5, ;6, 8, 72, ;68, , ;5, ;;, 45, B2, AA, A5, ;A, 8, 4B

2. hat is your story

ork individually. 'fter 46 minutes, share your answer to the person next to you.

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%he bar chart below does not have a title and other important information ismissing:

+se your imagination and knowledge of charts to help make sense of theabove chart. %hink of a suitable title that explains what the bar chart is all about.

Provide all the needed information and labels to complete the graph.

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c. 3rs. !ernardo tallied the econd -uarter 3ath grades of all 266 Grade ;

students in (aras school as follows:

B5 M 466 27 students

B6 M B 8 students

A5 M AB students

A4 M A 76 students;A M A6 25 students

; M ;; 46 students

;6 M ; 44 students

2. Go to your school $3= "$ducational 3anagement =nformation ystem# office and

ask permission from the personnel at the principals office to guide you where the

graphsEcharts of the Performance =ndicators of the school for the past 2 school

years.

a. hat are the performance indicators that you see

b. hat information can be obtained from the charts or graphsc. hat is the significance of those charts or graphs

d. hy is it important to put the data in graphs or in charts

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Lesson 3! r>ani?in> and +resentin> .ata @sin> Fre6(enc 4a)le andAisto>ram

+rere6(isite once'tsorting and Presenting &ata

*)o(t the Lesson%his lesson focuses on the use of freCuency tables and histograms as a way oforgani0ing and presenting data.

)8ectie=n this lesson, you are expected to:4. ort and organi0e data using freCuency table29 Present data using a histogram.

Lesson +ro'er

I9 *ctiit

%he following is a list of shoe si0es of 7 boys.; 8 8 5 8A 8 7 5 7 5 7 2 A 5 5 ; 8 ; 8 8 8 2 5 8

a. hat information can you get from the above listb. hat does the list tell usc. %he list above does not tell us anything. o what do we do with the list

II9 .isc(ssion

4. %he )reCuency %able

e construct a table with three columns as shown. %he shoe si0es may be grouped asfollows: 4 9 7, 7 9 , 9 ;, and ; 9 B. %hen we refer to our list and go down each columnand make a mark for each figure or number in the tally table. )or boundary si0es, we canuse the rule that we include the lower bound in the interval but not the upper bound. %hefirst count is 2 "for si0es 4 and 2#, so we put two strokes. %he next count is 42 "for si0es7 and 8#, so we put 42 strokes, and so on.

Shoe Si?e 4all Fre6(enc

4 9 7

7 9 9 ;; 9 B

OO

OOOO OOOO OO OOOO OOOO OOOO O OOOO

2

4245

4otal 3

'fter tallying, we count the tally marks in each row and put the number in the lastcolumn. %his is now called the freCuency table.

&"

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2. %he )reCuency Histogram

)rom the freCuency table, we are going to construct a histogram. ' histo>ramis arepresentation of a freCuency distribution by means of rectangles whose widthsrepresent class intervals and whose areas are proportional to the correspondingfreCuencies.

&! !# #' '%

#

&%

%

$isto%ram of &oys' hoe i(es

hoe i(es

)re*uency +ount

a. hat does the height of each of the rectangles tell usThe height of each rectangle corresponds to the frequency of the shoe sizeswithin an interval.

b. hat is the total area of all the rectanglesThe total area of the rectangles is equal to 35

c. hat does the total area tell usThe total area of the rectangles is the total frequency.

III9 *ctiitGiven the set of data, construct a freCuency table and a freCuency histogram usingintervals of width .

78 8 7; 2B 2675 26 26 78 886 86 78 8 8676 76 26 2B 75

%he lowest data value is 26 so we can use 5 intervals of width : 26 M 2, 2 M 76,76 M 7, 7 M 86, 86 M 8, and 8 M 6.

=nterval )reCuency26 M 2 82 M 76 276 M 7 7 M 86 786 M 8 78 M 6 7

%otal 26

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'nd here is the corresponding freCuency histogram made by $xcel:

%

&

!

"

#

$

,ntervals

)re*uency

*an you come up with a different set of intervals and freCuency count uppose youused 8 or intervals only How will the histogram differ

I

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hat is a freCuency histogramA histogram is a vertical bar graph of a frequency distribution of data values grouped into

intervals.

How do you construct a freCuency histogram

Step 1:,lace the data intervals along the horizontal a-is.

Step 2:ar* the frequency nubers on the vertical a-is.

Step 3:/raw rectangles for each interval. The height of the rectangle is the frequency count

for that interval.

ram is a vertical bar graph of a freCuencydistribution of data values grouped into intervals.

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Lesson %0 *era>es Mean, Median, and Mode

+rere6(isite once'tsorting and Presenting &ata

*)o(t the Lesson

%his is a lesson on computing for the values of 3ean, 3edian, and 3ode given a setof data. ou will also describe data using the mean, median, and mode. Problem solving

involving 3ean, 3edian, and 3ode is also included in this lesson.

)8ectie

=n this lesson, you are expected to:

4. )ind the mean, median, and mode of a statistical data.

2. olve problems involving mean, median, and mode.

7. &escribe the data using information from the mean, median, and mode.

Lesson +ro'er

I9 *ctiit

%he set of data shows a score of 7 students in their periodical test.

4. hat score is typical to the group of students hy2. hat score freCuently appears7. hat score appears to be in the middle How many students fall below this

score

II9 .isc(ssion

4. %he mean is commonly referred to as the average of all values. %o compute for the

mean, add all the scores and divide the sum by the number of scores. =t is the

easiest QaverageR to compute.

2. %he most freCuent scoreEs in the given set of data is called the mode. =t is also an

QaverageR score. ' data set may have two modes "and hence the data set is called

bimodal#.

7. %he median is also an QaverageR score. =t is the middle score in the list after the

scores are arranged in decreasing or increasing order. How do you get the median

for an even number of scores in a set of data hat about for an odd number of

scores

III9 *ctiit

&(

!" !# "% "% "( &

& % & !" "# & %

& &' &( &$ % (

& % &( &' &% "# "(

& &' "# #% "( #

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4. %he following sets of data show the height Kin centimetersL of two groups of boys

playing basketball.

Group ': 47, 475, 486, 46, 478, 42B, 425, 476Group !: 45;, 475, 44B, 475, 456, 4;A, 425, 486

a. *ompute for the mean.

b. hat information can you get from these two values

2. %he following sets of data show the weekly income Kin pesoL of ten selected

households living in two different barangays in the town of Sananga.

!rgy. Sawayan: 46, 466, 4;66, 4A66, 7666, 2466, 4;66, 466, 4;6,

4266

!rgy. 3ontealegre: 4666, 4266, 4266, 446, 4A66, 4A66, 4A66, 2666,48;6, A666

a. *ompute for the mean and the median for each !arangay.b. hat information can we get from these valuesc. hy do you think the median is more appropriate than the mean

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Lesson %1 *nal?in>, Inter'retin>, and .ra5in> oncl(sions /rom Gra'hicsand 4a)(lar +resentations

+rere6(isite once'ts1rgani0ing data using charts, tables and graphsD 3ean, 3edianand 3ode

*)o(t the Lesson%his lesson serves as a consolidation and practice of what you learned in the

previous lessons. %he problems will give you the opportunity to work in groups and discussdifferent solutions. ou will also learn to ustify your answers using data.

)8ectie=n this lesson, you are expected to analy0e, interpret accurately and draw

conclusions from graphics and tabular presentations of statistical data.

Lesson +ro'er

I9 *ctiit

olve the following problems in groups or in pairs.

4. &aria bought 7 colors of %9shirts from a department store. he paid an average ofPhP ;8.66 per shirt. %he receipt is shown below where part of it was torn.

2.

a. How much did she pay for each white shirtb. How much did she pay in all How did you determine this

&

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7. amuel brought ten sachets of chocolate candies. He checked the sachets andfound that they contain 42, 4, 45, 46, 4, 48, 42, 45, 4, and 42 candies.

a. 'ccording to the data, what is the mean number of candies per sachet

-ER-/E +012E123 #4

b. %he above information is written on each pack of candies. hy do you thinkthis number is different from the answer to "a#

8. %he 3unicipal 'griculturist of Sananga made a survey of fruit trees available in theorchard. He made a chart that shows the distribution of types of fruit trees planted.

ambuttan* "%+

-angosteen* %+

Durian* %+

0ackfruit* &%+

1antol* &%+

1o. of )ruit 2rees

a. How do the number of &urian and /ambutan trees compareb. hat is the most common fruit treec. hat fraction of the fruit trees is antold. =f there are 46 fruit trees altogether, how many are 3angosteen trees

II9 *ctiit

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4. 3rs. 'modia, the chool (ibrarian prepared a bar chart that shows the number ofstudents who borrowed maga0ines in the library last week.

-onday Tuesday Wednesday Thursday 2riday

%

#

&%

%

#

!%

1o. of Ma%a(ines &orroed

a. How many maga0ines were borrowed on )riday How many studentsborrowed maga0ines on this day

b. hat is the mean number of maga0ines borrowed per day last week Howmany students borrowed maga0ines in all of the five days

c. 1n what day had the most number of students borrow maga0inesd. How many students borrowed maga0ines on %uesday

2. %he pie chart below shows the memberships of the different clubs in 1rmoc *ityNational High chool.

!

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+lub Membership

a. hat is the ratio of the number of students who are members of the

*omputer *lub to that of the members of the Glee *lubb. hat percentage of the students are members of the &rama *lubc. =f there are a total 286 students, how many are members of the 3athematics

*lub

III9 :(estions;+oints to +onder

!y analy0ing data from graphs, charts and tables carefully, you could derive

important information that lead to well9ustified answers or conclusions. Hence it is important

for students what each type of data presentation emphasi0es. *ombine with that the

understanding of the measures mean, median and mode and the information gets richer.%his is the most important part of this lesson.

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7. +& to PHP $xchange /ates: "+ &ollar to Philippine Peso# *harts and historicaldata

Last 10 5orCin> das

a. 1n what day was the peso strongest against the + dollar

b. 1n what days did the peso appreciate against the + dollar

c. *an you explain the scale on the vertical axis

d. hat is the QaverageR peso9dollar exchange rate from &ec 28 to >an B