Seventh Grade Tasks Weekly Enrichment Teacher · PDF file · 2014-07-03SOLVING MATH...

$Page 1: Seventh Grade Tasks Weekly Enrichment Teacher · PDF file · 2014-07-03SOLVING MATH PROBLEMS ... fractions written as hundredths, the percent, or the decimal form to compare easily.$

Seventh Grade

Problem Solving Tasks‐ Weekly Enrichment

Teacher Materials

Summer Dreamers



By the end of this lesson, students should be able to answer these key questions:

How do you generate equivalent rational numbers?

How do you compare rational numbers?

MATERIALS:

Warm‐Up: Who is Correct?

Activity Master: Number Line – assembled and posted on the wall

For each student:

Fractions, Decimals, and Percents, Oh My!

Race Car Stat

Evaluate: Equivalent Rational Numbers

For each group of 2 students:

Activity Master: Fractions, Decimals, and Percents – cut apart, 1 set of

cards per group

SOLVING MATH PROBLEMS

KEY QUESTIONS

WEEK 1

TEACHER TOOLS

ENGAGE: The Engage portion of the lesson is designed to access students’ prior

knowledge of percent models. This phase of the lesson is designed for groups of 2

students. (10 minutes)

1. Distribute “Who is Correct?” warm‐up.

2. Prompt students to individually complete the warm‐up “Who is

Correct?”

3. Upon completion of the warm‐up, prompt students to share and justify

their solutions with a partner.

4. Actively monitor student work and ask facilitating questions when

appropriate.

Facilitating Questions:

What is the question asking you to do?

Answers may vary. Possible answer: Determine who is correct by determining the

percent of the flag that is shaded.

What do you know?

Answers may vary. Possible answer: I know the answer given by each person, and I

was given a picture of the flag.

What do you need to know?

Answers may vary. Possible answer: I need to know the percent of the flag that is

shaded in order to determine who is correct.

What strategy could you use to determine who is correct?

Answers may vary. Possible answer: I could find the percent of the flag that is shaded

then compare my answer to the answer of each person to determine who is correct.

How many squares make up the flag?

32

How many squares are shaded?

12

How could you write a ratio that compares the number of shaded squares

to the total number of squares on the flag?

Answers may vary. Possible answer: 3 to 8, 3:8, 3 out of 8, 3/8

What do you know about percents?

Answers may vary. Possible answer: I know that percents are how many out of a 100.

How could you use the ratio of the shaded squares to the total number of

squares to help you determine what percent of the flag is shaded?

Answers may vary. Possible answer: I know the ratio of shaded squares to total

squares is 3/8; therefore, I could use a factor of change to rewrite the ratio as a

fraction with a denominator that is a power of 10, such as 1000.

What factor of change could you use to change 3/8 to thousandths?

Multiply by 125

What is 3/8 written as thousandths?

375/100

How could you rewrite 375/1000 as a percent?

Answers may vary. Possible answer: Multiply the numerator and the denominator by

1/10 in order to generate an equivalent fraction with a denominator of 100, 37.5/100.

Then I could use the numerator as my percent, since percent means out of 100.

EXPLORE: The Explore portion of the lesson provides the student with an

opportunity to be actively involved in investigating equivalent rational numbers.

This phase of the lesson is designed for groups of 2 students. (25 minutes)

1. Distribute 1 set of Activity Master: Fractions, Decimals, and Percents to

each group of students and Fractions, Decimals, and Percents, Oh My! to

each student. (NOTE: Have Activity Master cards pre‐cut for student use.)

2. Prompt students to complete Fractions, Decimals, and Percents, Oh My!


appropriate.

Facilitating Questions:

What information is found on the cards?

Answers may vary. Possible answer: The cards contain rational numbers written in

different forms.

Rewriting Fractions as Decimals

How could rewriting each fraction as hundredths help you write the

decimal representation of the fraction?

Answers may vary. Possible answer: Decimals are just fractions that have

denominators that are 10, 100, 1000, etc. (powers of 10). So if I rewrite the fraction as

hundredths, I could write the decimal by using place value.

What factor of change could you use to rewrite this fraction as

hundredths?

Answers may vary.

Rewriting Fractions as Percents

How could rewriting each fraction as hundredths help you write the

percent representation of the fraction?

Answers may vary. Possible answer: Percents are just fractions that have

denominators of 100. So if I rewrite the fraction as hundredths, I could write the

percent by using the numerator.


hundredths?

Answers may vary.

x 125

. 62.5%

Rewriting Decimals as Percents

How could rewriting each decimal as a fraction help you write the decimal

as a percent?

Answers may vary. Possible answer: Place value is based on powers of 10: 10, 100,

1000, etc. So if I rewrite the decimal as a fraction, I could apply a factor of change to

the fractions to rewrite the fractions as hundredths then I could write the percent by

using the numerator.


hundredths?

Answers may vary.

Rewriting Decimals as Fractions

How could place value help you write each decimal as a fraction in

simplest form?

Answers may vary. Possible answer: Place value is based on powers of 10: 10, 100,

1000, etc. So I could rewrite the decimal as a fraction with a denominator of tenths,

hundredths, or thousandths then simplify.

Rewriting Percents as Fractions

What procedures could be used to write a percent as a fraction in

simplest form?


denominators of 100. So if I rewrite the fraction as hundredths, then I could simplify.

Rewriting Percents as Decimals

How could rewriting a percent as a fraction help you write a percent as a

decimal?


denominators of 100. So if I rewrite the percent as a fraction, I could write the decimal

by using place value.

Ordering from Least to Greatest

Which representation is the easiest to use to help you determine which

rational number represents the largest amount? Why?

Answers may vary. Possible answer: To compare the rational numbers, I could use the

fractions written as hundredths, the percent, or the decimal form to compare easily.


rational number represents the smallest amount? Why?



Comparing 83.5%

How could you determine which of the numbers are equivalent to 83.5%?

Answers may vary. Possible answer: I could rewrite 83.5% as a fraction and as a

decimal and compare my values with the values of the answer choices.

What process could you use to rewrite 83.5% as a fraction with a

denominator of 100?


denominators of 100. So I could rewrite 83.5% as a fraction where 83.5 is the

numerator and 100 is the denominator.

What process could you use to rewrite 83.5% as a fraction with a

denominator of 1000?

Answers may vary. Possible answer: I could rewrite 83.5% as a fraction where 83.5 is

the numerator and 100 is the denominator then use a factor of change to rewrite the

fraction as thousandths.

What factor of change could be used to convert 83.5/100 to thousandths?

10

What process could you use to rewrite 83.5% as a decimal?

Answers may vary. Possible answer: I could rewrite 83.5% as a fraction with a

denominator of 1000 then write the decimal by using place value. 0.835

EXPLAIN: The explain portion of the lesson provides students with an opportunity

to express their understanding of equivalent rational numbers. The teacher will

use this opportunity to clarify vocabulary terms and connect student experiences

in the Explore phase with relevant procedures and concepts. (20 minutes)

1. Display assembled Activity Master: Number Line on the wall in front of the

room.

2. Prompt 1 group of students with Card Set 1 to post their cards in the

appropriate place on Activity Master: Number Line. Students will need to

approximate the placement.

3. Prompt 1 group of students with Card Set 2 and 1 group with Card Set 3 to

add their cards to the number line.

Note: There are repeated rational numbers throughout the 3 different sets

of cards; however, the repeated rational numbers are in different forms.

4. Use the facilitating questions to lead a whole‐group discussion as students

add their cards to the number line.

Number Line Key

0.625

12.5% 20/32

1/8 1/5 20.4% ¼ 32.6% 5/8

0 0.5

Facilitating Questions





Which representation would be the hardest to use to determine which


Answers may vary. Possible answer: To compare the rational numbers, I would not

use the fractions in the simplest form because these fractions are not easily compared

without a common denominator.

How could you use the placement of the cards on the number line to

determine which rational number represents the largest amount?

Answers may vary. Possible answer: The number that represents the largest amount

would be the number that is farthest to the right on the number line.




fractions written as hundredths, the percent, or the decimal form to easily compare.

Which representation would be the hardest to use to determine which


Answers may vary. Possible answer: To compare the rational numbers, I would not

use the fractions in simplest form because they do not have a common denominator.

How could you use the placement of the cards on the number line to

determine which rational number represents the smallest amount?

Answers may vary. Possible answer: The number that represents the smallest amount

would be the number that is farthest to the left on the number line.

Which representation is the easiest to use to help you determine where a

rational number lies on the number line? Why?

Answers may vary. Possible answer: Since the number line is in decimal form, it is

easier to determine the placement of the decimal representations.

5. Debrief questions 3 on Fractions, Decimals, and Percents, Oh My!

6. Use the facilitating questions to lead the discussion.


How did you determine which of the numbers are not equivalent to

83.5%?

Answers may vary. Possible answer: I rewrote 83.5% as a fraction and as a decimal

and compared my values with the values of the answer choices.

Did you eliminate any of the answer choices? Why?

Answers may vary. Possible answer: Yes, since 83.5% is less than 100% and 100% is

equivalent to 1, I was able to eliminate 8.35 because it is greater than 1.

What process did you use to rewrite 83.5% as a fraction with a

denominator of 100?


denominators of 100. So I rewrote 83.5% as a fraction where 83.5 is the numerator

and 100 is the denominator.

What process did you use to rewrite 83.5% as a fraction with a

denominator of 1000?

Answers may vary. Possible answer: I rewrote 83.5% as a fraction where 83.5 is the

numerator and 100 is the denominator then used a factor of change of 10 to rewrite as

thousandths. (835/1000)

What process did you use to rewrite 83.5% as a decimal?

Answers may vary. Possible answer: I rewrote 83.5% as a fraction with a

denominator of 1000 then wrote the decimal by using place value.

Which number is not equivalent to 83.5%? Why?

8.35

ELABORATE: The Elaborate portion of the lesson affords students the opportunity

to extend or solidify their knowledge of equivalent rational numbers. This phase

of the lesson is designed for individual investigation. (10 minutes)

1. Distribute Race Car Stat to each student.

2. Prompt students to complete Race Car Stat. (Answer A)


appropriate.



Answers may vary. Possible answer: Determine between which 2 fractions 3/8 lies on

a number line.

What do you know?

Answers may vary. Possible answer: I know 4 different possible sets of fractions that

3/8 may fall between.

What do you need to know?

Answers may vary. Possible answer: I need to know a common denominator so that I

can make comparisons.

What procedure could you use to determine which pair of fractions 3/8

may fall between?

Answers may vary. Possible answer: I could find a common denominator, simplify, or

use a factor of change to rewrite each fraction using the common denominator then

compare numerators.

EVALUATE: During the Evaluate portion of the lesson, the teacher will assess

student learning about the concepts and procedures that the class investigated

and developed during the lesson. (20 minutes)

1. Distribute Evaluate: Equivalent Rational Numbers to each student.

2. Prompt students to complete Evaluate: Equivalent Rational Numbers.

3. Upon completion of Evaluate: Equivalent Rational Numbers, the teacher

should discuss error analysis (shown below)to assess student

understanding of the concepts and procedures the class addressed in the

lesson.

Answers and Error Analysis for Evaluate: Equivalent Rational Numbers

Question Number

Correct Answer

Conceptual Error Procedural Error

1 B A C D

2 B A C D

3 A B C D

4 A B C D

STUDENT WORKSHEETS FOLLOW!!!!!

Warm‐Up: Who is Correct?

Kobie shaded 3/8 of the flag black, as shown below.

Cassie stated that 12% of the flag was shaded, and Kobie said that 37.5% of the

flag was shaded. Who is correct? Explain your answer.

Activity Master: Number Line

1

0.5

Activity Master: Number Line

0

Activity Master: Fractions, Decimals, and Percents

Cut each card out along lines. 1 set per group of students – 3 cards per set.

20.4%

Set 1 Card 1

Set 1 Card 2

Set 1 Card 3

0.625

Set 2 Card 1

Set 2 Card 2

Set 2 Card 3

Activity Master: Fractions, Decimals, and Percents

Set 3 Card 1

Set 3 Card 2

Set 1 Card 3

Name: ______________________________________ Date:_________________

Fractions, Decimals, and Percents, Oh My!

Complete the table below.

CARD Fraction (in simplest form)

Decimal Percent

Card 1

Card 2

Card 3

1. List the cards in order from least to greatest.

2. Which representation is the easiest to use to help you determine the order from least

to greatest? Why?

3. It is estimated that Jimmy Johnson completed 83.5% of the laps in the 2004 Talladego

Race. Which number is NOT equivalent to 83.5%?

A.

B. .

C. 0.835

D. 8.35

Name: _______________________________________Date: ________________

Race Car Stat

Tony Stewart was either the winner or the runner‐up in 3 out of the last 8 races

in the series. The fraction 3/8 is found between which pair of fractions on a

number line?

A. and

B. and

C. and

D. and

Justify your answer choice and state why the other answer choices are

incorrect.

Name: ______________________________________Date:________________

Evaluate: Equivalent Rational Numbers

1. The fraction is found between which pair of fractions on the number line?

A. and

B. and

C. and

D. and

2. A specialty paint shop had 4 different race cars to complete. The shop completed , ,

, and of the work on each car. Which list shows the percent of the work completed

on each car in order from greatest to least?

A. 50%, 62.5%, 75%, 20%

B. 75%, 62.5%, 50%, 20%

C. 0.75%, 0.625%, 0.5%, 0.2%

D. 20%, 50%, 62.5%, 75%

3. Tyler estimated that 48.2% of the crystals in his sugar project developed correctly.

Which number is NOT equivalent to 48.2%?

A. 4.82

B. 0.482

C.

D. .

4. The table shows the driver and the portion of allowable gas each driver used in the race.

Gas Usage

Driver Portion of Allowable Gas

Used

Busch

Johnson

Burton 48.3%

Earnhardt

Harvick 48.2%

Which of the following lists the racers in order from least to greatest portion of

allowable gas used?

A. Busch, Earnhardt, Harvick, Burton, Johnson

B. Busch, Earnhardt, Burton, Harvick, Johnson

C. Johnson, Burton, Harvick, Busch, Earnhardt

D. Johnson, Burton, Harvick, Earnhardt, Busch



What is a ratio?

What are the different ways to write a ratio?

How can you determine if 2 ratios are equivalent?

What is a proportion?

How can you determine if a problem situation can be solved using a

proportion?

What are the different ways to write a proportion given a problem

situation?

MATERIALS:

For each student:

Warm‐Up Activity: Find Someone Who…

Heartbeats

Speed Racer

Evaluate: Ratios and Proportions

For each group of 2 to 3 students:

Activity Master: Spinners – cut apart – 1 set per group

Paperclip – 1 per group


KEY QUESTIONS

WEEK 2

TEACHER TOOLS


knowledge about ratios. This phase of the lesson is designed for whole‐group

instruction. (15 minutes)

1. Distribute a Formula Chart (optional) and Find Someone Who… to each

student.

2. Prompt students to complete and Find Someone Who…


appropriate.


What is a ratio?

Answers may vary. Possible answer: A ratio is a comparison of two or more numbers.

Does order matter when writing a ratio? Why?

Answers may vary. Possible answer: Yes, the relationship must stay the same;

however, if I label my values, then order does not matter.

What are the different ways to write a ratio?

Answers may vary. Possible answer: Different ways to write a ratio include: using a

colon, as a fraction, verbally using the word “to” or using the words “out of.”

How could you simplify a ratio?

Answers may vary. Possible answer: You can simplify a ratio by dividing each of the

numbers in the ratio by the same factor.

Where could you find the relationship between seconds and minutes?

Answers may vary. Possible answer: Formula Chart

What is the relationship between seconds and minutes?

60 seconds = 1 minute

How could you use this relationship when finding the number of

heartbeats in a minute?

Answers may vary. Possible answer: Since I know there are 60 seconds in a minute

and 10 seconds goes into 60 seconds 6 times, I could multiply the number of

heartbeats in 10 seconds by 6.


opportunity to be actively involved in investigating ratios and proportions. This

phase of the lesson is designed for groups of 2 to 3 students. (20 minutes)

1. Distribute Heartbeats to each student.

2. Distribute 1 set of Activity Master: Spinners and a paperclip to each group

of students. (Use a pencil with the paperclip to spin the paperclip.)

3. Prompt students to complete Heartbeats.

4. Actively monitor student work and ask facilitating questions.


What ratio of seconds to heartbeats did you get when you used the 2

spinners?

Answers may vary.

Is it possible to simplify this ratio?

Answers may vary.

Which time value(s) are multiples of the time interval you spun?

Answers may vary. Possible answer: Since I spun 30 seconds, 60 seconds is a multiple

of 30 seconds.

Which time value(s) are factors of the time interval you spun?

Answers may vary. Possible answer: Since I spun 30 seconds, 15 seconds is a factor of

30 seconds.

What factor could be used to scale your time interval up or down?

Answers may vary. Possible answer: Since I spun 30 seconds, I could use a factor of ½

to scale down to 15 seconds, a factor of 1 ½ to scale up to 45 seconds, and a factor of 2

to scale up to 60 seconds.

How could you use this factor to determine the number of heartbeats?

Answers may vary. Possible answer: Since I spun 30 seconds and 8 heartbeats, I could

use a factor of ½ to scale down the number of seconds to 15 and the number of

heartbeats to 4 heartbeats.

What is the relationship between seconds and minutes?

60 seconds = 1 minute

If you know the number of heartbeats in 30 seconds, how would you

determine the number of heartbeats in 1 minute?

Answers may vary. Possible answer: Since I could multiply 30 seconds by a factor of 2

to get 60, I could multiply the number of heartbeats by 2.

EXPLAIN: The Explain portion of the lesson provides students with an opportunity

to express their understanding of ratios and proportions. The teacher will use this

opportunity to clarify vocabulary and connect student experiences in the Explore

phase with relevant procedures and concepts. (15 minutes)

1. Debrief Heartbeats.



What is a ratio?

Answers may vary. Possible answer: A ratio is a comparison of two values.

What are some was ratios can be recorded?

Answers may vary. Possible answer: Ratios can be recorded as a fraction, with a

colon, or with the words “to” or “out of.”

What patterns did you see in the table?

Answers may vary. Possible answer: The number of heartbeats is always a multiple of

16.

How did you determine the number of heartbeats in 1 minute?

Answers may vary. Possible answer: Since I know that 1 minutes is 60 seconds, I

continued the pattern in the table until I found the number of heartbeats in 60

seconds. I set up and solved a proportion.


Answers may vary. Possible answer: A proportion is an equation showing that two

ratios are equivalent.

How do you know if a proportion could be used to solve these problems?

Answers may vary. Possible answer: A proportion may be used if the problem contains

a ratio and the situation requires the ratio to be scaled up or down.

How could you set up a proportion to solve this problem?

Answers may vary. Possible answer:

= =

What process could you use to find the missing value in your proportion?

Answers may vary. Possible answer: Since 60 is a multiple of 15 and 4 times 15 equals

60, then I could multiply 16 times 4.

How many times would the heart beat in 1 minute?

Answers may vary. Possible answer: 64 times.

How do you determine the number of heartbeats in 3 minutes?

Answers may vary. Possible answer: I multiplied the number of heartbeats for 1

minute by 3.

How many times would the heart beat in 3 minutes?

Answers may vary. Possible answer: 192 times.

How did you determine the number of seconds for 240 heartbeats?

Answers may vary. Possible answer: I set up and solved a proportion.

How could you set up a proportion to determine the number of seconds

for 240 heartbeats?


= =

What process could you use to find the missing value in your proportion?

Answers may vary. Possible answer: Since 16 times 15 equals 240, I could multiply 15

times 15.

How many seconds will pass for the heart to beat 240 times?

Answers may vary. Possible answer: 225 seconds.

How could you find the number of minutes for 240 heartbeats?

Answers may vary. Possible answer: Since I know that 60 seconds is 1 minute, I could

divide 225 by 60.

What process did you use to determine how many times Carol’s heart

beat while walking for 4 minutes?

Answers may vary. Possible answer: Since her heart beats 18 times in 10 seconds, I

found how many times her heart would beat in 1 minute then multiplied that number

by 4 to find the number of heartbeats in 4 minutes.

How could you set up a proportion to solve this problem?


= =

What factor could you use to find the missing value in your proportion?

Answers may vary. Possible answer: Since 24 times 10 equals 240, I multiplied 18

times 24.

About how many times would Carol’s heart beat during 4 minutes of

walking?

432 times.


to extend or solidify their knowledge of ratios and proportions. This phase of the

lesson is designed for individual investigation. (15 minutes)

1. Distribute Speed Racer to each student.

2. Prompt students to complete Speed Racer.


appropriate.



Answers may vary. Possible answer: Determine which girl answered the question

correctly.

What information is given to you?

Answers may vary. Possible answer: We know each girl’s answer choice and the

problem they solved.

What strategy could be used to determine which girl is correct?

Answers may vary. Possible answer: I could solve the problem and then compare my

answer to the answers of Maria and Louisa.

What ratio is described in thee problem situation?

Answers may vary. Possible answer: 85 miles/1 hour

What proportion describes the situation?

Answers may vary. Possible answer: = =

Which verbal description matches the process you could use to find the

missing value in the proportion?

Answers may vary. Possible answer: Since I would need to determine the factor to use

to scale up 85 to 255, I could divide 255 by 85. Therefore, answer choice A has the

correct verbal description.



and developed during the lesson.(20 minutes)

1. Distribute Evaluate: Ratios and Proportions to each student.

2. Prompt students to complete Evaluate: Ratios and Proportions.

3. Upon completion of Evaluate: Ratios and Proportions, the teacher should

use the error analysis, provided below, to assess student understanding of

the concepts and procedures the class addressed in the lesson.


Question Number

Correct Answer


1 B A C D

2 C A B D

3 24

4 C A D B

STUDENT WORKSHEETS FOLLOW!!!!!


Name: ____________________________________Date: ____________________

Name: ____________________________________Date: ____________________

Name: ____________________________________Date: ____________________

Name: ____________________________________Date: ____________________

Evaluate: Ratios and Proportions

1. If the ratio of cats to dogs in the veterinarian clinic is 2 to 3, which ratio

does NOT show a possible number of cats to dogs in the clinic?

A. 36 cats, 54 dogs

B. 34 cats, 21 dogs

C. 24 cats, 36 dogs

D. 30 cats, 45 dogs

2. Claire was making necklaces for the craft show. She completed 9 necklaces

in 30 minutes. If Claire continued making necklaces at this rate, how many

necklaces would she make in 2 hours?

A. 4

B. 9

C. 36

D. 45

3. The ratio of butterflies to bees in Jane’s insect collection is 3 to 4. If there

were 32 bees, how many butterflies would be there? Show all work and

explain your reasoning.

4. There were 16 box cars and 24 students registered for the box car

tournament. Which ratio accurately compares the number of students to

the number of box cars?

A. 2: 12

B. 3: 1

C. 3: 2

D. 16: 24





How can you represent a proportional relationship by using a ruler?

What patterns are there in proportional relationships?

How do you solve proportions?

MATERIALS:

For each student:

Round Robin Warm‐Up Activity: Proportions

School Project

Paintbrushes

Evaluate: Proportions


KEY QUESTIONS

WEEK 3

TEACHER TOOLS


knowledge of comparing rates. This phase of the lesson is designed for groups of

4 students. (15 minutes)

1. Distribute Round Robin: Proportions warm‐up activity to each student.

2. Prompt students to complete Round Robin: Proportions warm‐up activity

with their partners.

3. Actively monitor student work and ask facilitation questions when

appropriate.


How could you determine the number of 10‐ounce bottles needed?

Answers may vary. Possible answer: I could divide 60 ounces, the total amount of red

paint needed, by 10 ounces.

How could you determine the total cost of the 10‐ounce bottles?

Answers may vary. Possible answer: I could multiply $1.60 by the number of 10‐ounce

bottles needed.

Does your answer seem reasonable? Why or why not?

Answers may vary. Possible answer: Yes. If I round $1.60 up to $2.00 and multiply it by

6, the number of 10‐ounce bottles needed, I get $12. My actual answer was $9.60.

Since I rounded the price per bottle up, I know the actual answer will be less than my

estimate.

How could you determine the number of 15‐ounce bottles needed?

Answers may vary. Possible answer: I could divide 60 ounces, the total amount of red

paint needed, by 15 ounces.

How could you determine the total cost of the 15‐ounce bottles?

Answers may vary. Possible answer: I could multiply $2.20 by the number of 15‐ounce

bottles needed.

How could you use the total cost of the 10‐ounce bottles and the 15‐

ounce bottles to determine the better buy?

Answers may vary. Possible answer: I could compare the total costs and see which

one is less to determine the better buy.


opportunity to be actively involved in using a ruler to model proportional

relationships. This phase of the lesson is designed for groups of 2 students. (25

minutes)

1. Distribute School Project to each student.

2. Prompt students to complete School Project with their partners.

3. Actively monitor students and ask facilitating questions when appropriate.


Part 1

If 2 inches represents 15 feet, how many feet are represented by 4

inches?

30 feet

How could you use the ruler to determine the number of feet on the

mural?

Answers may vary. Possible answer: Every 2 inches represents 15 feet, so I could mark

an additional 15 feet every time I increase by 2 inches on the ruler.

How could you use the fact that 3 inches is halfway between 2 inches and

4 inches on the ruler to determine the number of feet represented by 3

inches?

Answers may vary. Possible answer: Since 3 inches is halfway between 2 inches and 4

inches, the number of feet must be halfway between 15 feet and 30 feet. I can find the

average of 15 feet and 30 feet.

What labels do you have on your ruler that could be used to determine

the number of inches that represent 135 feet?

Answers may vary. Possible answer: Since 8 inches represents 60 feet and 10 inches

represents 75 feet, I can add those 2 values together to see that 18 inches are needed

to represent 135 feet.

Part 2

Since 12 centimeters on the ruler is divided into 3 equal sections, how

many cups of white paint does each section represent? How do you

know?

÷ 3 =

What patterns do you notice on the ruler?

Answers may vary. Possible answer: Every increase in 4 centimeters (blue paint)

corresponds to an increase in cup of white paint.

Part 3

How could you label the ruler to help you solve this problem?

Answers may vary. Possible answer: I could label the top of the ruler as silver stars

and the bottom of the ruler as gold stars. Since there are 8 silver stars for every 3 gold

stars, I could put an 8 above the 3‐inch mark on the ruler.

What other value(s) could you find on the ruler? How?

Answers may vary. Possible answer: I could find the number of silver stars

represented by 6, 9, and 12 gold stars by using the 6‐inch, 9‐inch, and 12‐inch marks on

the ruler.

What patterns do you notice on the ruler?

Answers may vary. Possible answer: Every time I increase 3 inches (the number of

gold stars), I increase by 8 silver stars.

EXPLAIN: The Explain portion of the lesson provides students with an opportunity

to express their understanding of proportional relationships. The teacher will use

this opportunity to clarify vocabulary terms and connect student experiences in

the Explore phase with relevant procedures and concepts. (15 minutes)

1. Debrief School Project.



Part 1

How did you use the ruler to determine the number of feet on the mural?

Answers may vary. Possible answer: Every 2 inches represented 15 feet, so I increased

by an additional 15 feet every time I increased by 2 inches on the ruler.

How did you use the ruler to determine the number of feet that 3 inches

on the scale drawing would represent?

Answers may vary. Possible answer: Since 3 inches is halfway between 2 inches and 4

inches, the number of feet must be halfway between 15 feet and 30 feet. I found the

average of 15 feet and 30 feet which is 22.5 feet.

What proportion could you use to determine the number of feet

represented by 3 inches?

Answers may vary. Possible answer: =

How do you “see” this proportion on the ruler?

Answers may vary. Possible answer: The 15 is above the 2 on the ruler, just like it is in

the proportion. The unknown is above the 3 on the ruler, just like it is in the

proportion.

How could you solve the proportion = to verify the answer you

found with the ruler?

Answers may vary. Possible answer: Since the scale factor is , I could multiply 15 by

to verify the answer I found with the ruler.

How did you use the ruler to determine the number of inches on the scale

drawing that would represent 135 feet on the mural?

Answers may vary. Possible answer: Since 8 inches represents 60 feet and 10 inches

represents 75 feet, I added those 2 values together to see that 18 inches would

represent 135 feet.

Part 2

How did you use the ruler to fill in the missing blanks?

Answers may vary. Possible answer: I noticed that 12 centimeters on the ruler was

divided into 3 equal sections, so I divided of a cup into 3 equal parts.

How did you determine the number of cups of white paint needed for 16

cups of blue paint?

Answers may vary. Possible answer: Since every 4 centimeters represents cup of

white paint, I know that cup of white is used for 12 cups of blue paint. Therefore, I

added cup to cup to determine that 1 cup of white paint would be needed.

How did you use the ruler to determine the number of cups of blue paint

that are needed for 3 cups of white paint?

Answers may vary. Possible answer: Since 1 cup of white paint is used for 16 cups of

blue paint, I know that 3 cups of white paint are used for 48 cups of blue paint. I know

that cup of white paint is used for 12 cups of blue paint. Therefore, I added 48 cups

and 12 cups to find that I would need 60 cups of blue paint.

Part 3

How did you use the ruler to represent the situation?

Answers may vary. Possible answer: I labeled the top of the ruler as silver stars and

the bottom of the ruler as gold stars. Since there are 8 silver stars for every 3 gold

stars, I put an 8 above the 3‐inch mark on the ruler.

How did you determine the number of silver stars needed for 9 gold

stars?

Answers may vary. Possible answer: Every time I increased by 3 inches (3 gold stars),

I also increased by 8 silver stars. So 6 inches represented 16 silver stars and 9 inches

represented 24 silver stars.

What proportion could you use to determine the number of silver stars

needed for 9 gold stars?

Answers may vary. Possible answer: =

How do you “see” this proportion on the ruler?

Answers may vary. Possible answer: The 8 is above the 3 on the ruler, just like it is in

the proportion. The unknown is above the 9 on the ruler, just like it is on the

proportion.

How could you solve the proportion = to verify the answer you found

with the ruler?

Answers may vary. Possible answer: Since the scale factor is 3, I could multiply 8 by 3

to verify the answer I found with the ruler.

How did you find the number of gold stars represented by 64 silver stars?

Answers may vary. Possible answer: The ruler only went up to 12 inches so the most I

could represent was 12 gold stars and 32 silver stars. Since I needed to find the

number of gold stars for 64 silver stars, I just pretended there were 2 rulers. There

would be a total of 24 inches which would represent 24 gold stars and 64 silver stars.

Summary

How does a ruler help you visualize a proportional relationship?

Answers may vary. Possible answer: It helps me organize my information and look

for patterns. It is similar to creating a table to represent a proportional relationship.

How is using a ruler similar to writing a proportion?

Answers may vary. Possible answer: The information is organized in a similar

manner. I can “see” the proportion when I set up the ruler.


to extend or solidify their knowledge of proportional relationships. This phase of

the lesson is designed for individual investigation. (10 minutes)

1. Distribute Paintbrushes to each student.

2. Prompt students to complete Paintbrushes.


appropriate.


How could you determine the cost of 1 paintbrush in a 3‐pack of

paintbrushes?

I could divide $2.25 by 3.

How could you determine the cost of 1 paintbrush in a 5‐pack of

paintbrushes?

Answers may vary. Possible answer: Once I know the price of 1 paintbrush in the 3‐

pack, I could subtract 5₵ from that price to find the price of 1 paintbrush in the 5‐pack.

How could you use the price of 1 paintbrush in a 5‐pack to determine the

cost of the 5‐pack of paintbrushes?

Answers may vary. Possible answer: I could multiply the price of 1 paintbrush by 5.



and developed during the lesson.

1. Distribute Evaluate: Proportions to each student.

2. Prompt students to complete Evaluate: Proportions.

3. Upon completion of Evaluate: Proportions, the teacher should use the

error analysis to assess student understanding of the concepts and

procedures the class addressed in the lesson.


Question Number

Correct Answer


1 18

2 C A B D(Guess)

3 D A B C

4 B A C D

STUDENT WORKSHEETS FOLLOW!!!!

Name: ____________________________________Date: ____________________

Name: ____________________________________Date: ____________________


Name: ____________________________________Date: ____________________

Name: ____________________________________Date: ____________________

Evaluate: Proportions

1. A 12‐ounce bottle of prescription lotion to treat poison ivy costs $18. If the

price per ounce is the same, how many ounces would be in a bottle that

costs $27? Show all work and explain your answer below.

2. George is on the track team and can run 50 meters in 8 seconds. If he runs

at the same rate, about how many seconds will it take him to reach the

375‐meter mark?

A. 7.5 seconds

B. 47 seconds

C. 60 seconds

D. 68 seconds

3. A 1‐liter bottle of soda costs 99₵ and a 2‐liter bottle of soda costs $1.59.

Which of these statements will help a shopper decide which bottle is the

better buy?

A. The 1‐liter bottle of soda is the better buy because it is less expensive

per liter.

B. The 1‐liter bottle of soda is the better buy because 99₵ is about $1, and

$1 goes into $1.59 about 1.5 times.

C. The 2‐liter bottle of soda is the better buy because it is more expensive.

D. The 2‐liter bottle of soda is the better buy because it is less expensive

than two of the 1‐liter bottles of soda.

4. The prices of 4 different bottles of lotion are given in the table.

Lotion Prices

Bottle Size(ounces)

Price(dollars)

25 $4.10

15 $2.43

10 $1.64

5 $0.86

Which size bottle of lotion has the lowest price per ounce?

A. The 25‐ounce bottle

B. The 15‐ounce bottle

C. The 10‐ounce bottle

D. The 5‐ounce bottle



What is a rate?

What is a unit rate?

How can knowing the unit rate help you to complete a pattern?

How can you determine if a problem situation can be solved using a unit

rate?

MATERIALS:

For whole class demonstration:

Timer or watch with a second hand

For each student:

Rulers (for straightedge)

Warm‐Up Activity: The Name Game

Word Power Worksheet

Independent Practice: Rush Hour

Rates to Tables, to Graphs

Evaluate: Rates


KEY QUESTIONS

WEEK 4


TEACHER TOOL

ENGAGE: The Engage portion of the lesson is designed to create student interest

in the concepts addressed. This part of the lesson is designed for whole group

instruction. (10 minutes)

1. Distribute The Name Game warm‐up activity to each student.

2. Using a timer or a watch with a second hand, prompt students to begin

printing their first name as many times as possible in 15 seconds.

3. Prompt students to record this information in the table on The Name Game

in the appropriate row. (NOTE: Students should record fractional parts of

their name if they were unable to completely write their full name when

time is called.)

4. Prompt students to use their recorded data to complete The Name Game.


appropriate.


How would you find the fractional part of your name?

Answers may vary. Possible answer: I can count how many letters are in my name –

that will be the denominator of my fraction. The number of letters I was able to write

will be the numerator.

How do you multiply fractions?

Answers may vary. Possible answer: Change any mixed numbers to improper

fractions. Then multiply the numerators together and multiply the denominators

together.

How can you find the number of names you would be able to write in 5

seconds?

Answers may vary. Possible answer: I can divide the number of names I wrote in 15

seconds by 3.

How do you divide fractions?

Answers may vary. Possible answer: I can multiply the first fraction by the reciprocal

of the second fraction.

How could you use the first row of your table to find the number of times

you could write your name in 60 seconds?

Answers may vary. Possible answer: Since 60 seconds is four times 15 seconds, I can

multiply the number of names in the first row by 4.

What is a strategy you can use to find the number of times you could

write your name in 105 seconds?

Answers may vary. Possible answer: I can add together the answers I got for 45

seconds and 60 seconds.


opportunity to be actively involved in the exploration of the mathematical

concepts addressed. This part of the lesson is designed for groups of 2 students.

(20 minutes)

1. Distribute Word Power to each student.

2. Prompt students to complete Word Power.


appropriate.


What is different about the two rates?

Answers may vary. Possible answer: Amy’s rate is expressed in words per minutes and

Yvonne’s rate is expressed in words per second.

How are the tables different?

Answers may vary. Possible answer: Yvonne’s table uses seconds and Amy’s table uses

minutes.

What values can you put in your table for the time periods?

Answers may vary. Possible answer: I can put any time period in the table and then

determine the number of words that can be typed in that amount of time. Since

Yvonne’s rate was 11 words in 15 seconds, I thought it would be easiest to use time

periods that are multiples of 15 seconds. I put 11 words in 15 seconds as the first row

of the table and then extended the table from there.

How many seconds are in 1 minute?

60 seconds

What fraction of a minute is 15 seconds?

of a minute

How can you determine the number of words Yvonne can type in 2

minutes?

Answers may vary. Possible answer: I can multiply 11 words by 4 to find the number

of words Yvonne can type in 1 minute. Then I can multiply 44 words by 2.5 to find the

number of words Yvonne can type in 2 minutes.

How can you determine the number of words Amy can type in 1 minute?

In 15 seconds?

Answers may vary. Possible answer: I can divide 130 words by 2 to find how many

words Amy can type in 1 minute. Since 2 minutes equals 150 seconds, I can divide

130 words by 10 to find how many words Amy can type in 15 seconds.

EXPLAIN: The Explain portion of the lesson is directed by the teacher to allow the

students to formalize their understanding of the concepts addressed in the lesson.

(15 minutes)

1. Debrief Word Power.

2. Use facilitating questions to lead the discussion.


What is a rate?

Answers may vary. Possible answer: A rate is a ratio that compares 2 different units.

What 2 units were compared on Word Power?

Words and time (seconds or minutes)

How did you determine the values to place in your table?

Answers may vary. Possible answer: I used an increment of time that was a multiple or

factor of the time period I was given. I could have used any time periods, but it was

easiest to use factors or multiples.


What could be done to find Yvonne’s typing rate in words per minute?

Answers may vary. Possible answer: Since Yvonne’s typing rate is given in words and

seconds, I can find how many words she can type in 60 seconds. Thhat would be the

same as how many words she can type in 1 minute.

=

= 44 words per minute

What could be done to find Amy’s typing rate in words per minute?

Answers may vary. Possible answer: I can divide the numerator and denominator by

2 to find the unit rate.

=


How can you use a unit rate to determine the fastest typist?

Answers may vary. Possible answer: I can see who types the most words in 1 minute.

What was Kathy’s reading rate?

or

= 6 words per second

How could you find Kathy’s reading rate in words per minute?

Answers may vary. Possible answer: Since Kathy’s reading rate is given in words and

seconds, I can find how many words she can read in 60 seconds. That would be the

same as how many words she can read in 1 minute.

=


What was Jamie’s reading rate?

=

= words per second

How could you find Jamie’s reading rate in words per minute?

Answers may vary. Possible answer: Since 15 is a factor of 60, I could find how many

words Jamie reads in 15 seconds and use that information to find how many words she

can read in 60 seconds.

=

=


What was Betsy’s readying rate?

How could you find Betsy’s reading rate in words per minute?

Answers may vary. Possible answer: Find the unit rate by dividing the numerator and

denominator by .

=


How could you find Betsy’s reading rate in words per second?

Answers may vary. Possible answer: Find the unit rate by dividing the numerator and

denominator by 90 since there are 90 seconds in minutes.

=

=

or words per second

How could you use the graph to determine the fastest typist/reader?

Answers may vary. Possible answer: Look at one time period and read the graph to see

who typed (read) the most words for that time period.

ELABORATE: The Elaborate portion of the lesson provides an opportunity for the

student to apply the concepts of the concept within a new situation. This part of

the lesson is designed for individual investigation. (20 minutes)

1. Distribute Independent Practice: Rush Hour to each student.

2. Prompt student to complete Independent Practice: Rush Hour. The

teacher may choose to use several of these problems as guided practice to

check for student understanding of the concepts and procedures addressed

in the lesson.


appropriate.



What ratio can be written to represent the relationship?

Answers may vary.

What is the unknown in this problem?

Answers may vary.

Does your answer seem reasonable?

Answers may vary.

Could you solve this problem another way? How?

Answers may vary.

Does the answer have a unit of measure? If so, what is it?

Answers may vary.



and developed during the lesson. (15 minutes)

1. Distribute Evaluate: Rates to each student.

2. Prompt students to complete Evaluate: Rates.

3. Upon completion of Evaluate: Rates, the teacher should use the error

analysis to assess student understanding of the concepts and procedures

the class addressed in the lesson.

Answers and Error Analysis for Evaluate: Rates

Question Number

Correct Answer


1 D A B C

2 A B C D (Guess)

3 D A B C

4 B C D A (Guess)

STUDENT WORKSHEETS FOLLOW!!!!

Name: ____________________________________Date: ____________________

The Name Game

1. Your teacher is going to time you for 15 seconds. Your job is to print your

first name as many times as you can during those 15 seconds. Record the

total number of times you were able to write your name and any

fractional parts of your name in the appropriate row of the table below.

2. Use the data you recorded to complete the rest of the table.

Time (in seconds)

Number of names you wrote

15

30

45

60

75

90

105

120

3. What patterns do you see in the table?

4. How many times do you think you could print your name in 5 minutes?

Justify your answer.

5. How many names could you have written in 5 seconds?

Name: ____________________________________Date: ____________________

Word Power

1. Yvonne types 11 words in 15 seconds. Amy types 130 words in 2

minutes. Use the tables and graph below to determine who types at the

faster rate of speed. Justify your answers.

Yvonne’s Typing Rate Amy’s Typing Rate

Seconds Words Minutes Words

15 11 2.5 130

Typing Rates

2. Kathy, Jamie, and Betsy took a speed reading course. Kathy read 180

words in 30 seconds. Jamie read 210 words in 45 seconds. Betsy read 480

words in 1 minutes. Use the tables and graph below to explore their

reading rates. Then arrange the readers in order from the slowest reader

to the fastest reader. Justify your answer.

Kathy’s Rate Jamie’s Rate Betsy’s Rate

Seconds Words Seconds Words Seconds Words

30 180 45 210 90 480

Reading Rates

Name: ____________________________________Date: ____________________

Independent Practice: Rush Hour

A rate is a ratio that compares 2 different units. Examples of rates include feet

per second, miles per gallon, and words per minute.

You can determine equivalent rates and create graphs and tables in order to

make predictions and comparisons.

Practice: During morning rush hour, Paul drove 5 miles in 15 minutes. How

many miles per hour was Paul driving at that time?

1. What is Paul’s driving rate?

2. How many miles per hour was Paul driving at that time?

3. Create a table to show Paul’s driving rate up to 75 minutes.


4. Use your table to create a graph showing Paul’s driving rate.

Name: ____________________________________Date: ____________________

Rates to Tables, to Graphs

1. A cookie recipe calls for 1 cups of sugar to make 18 cookies. Complete

the table to determine the number of cups of sugar needed to make 24

cookies.

Cups of Sugar Cookies

1

18

2. Which car gets better gas mileage: one that can travel 150 miles on 6

gallons of gas or one that can travel 96 miles on 4 gallons of gas? Justify

your answer.

3. The copy machine makes 20 copies in 45 seconds. Complete the table

to determine the number of seconds it will take to make 640 copies.

Seconds Copies

45 20

4. Drew is paid $45 for 6 hours at his lifeguard job. At this rate, how much

will he be paid for 35 hours of work?

5. Nancy’s car travels 300 miles on 12 gallons of gas. How many gallons of

gas will she need in order to drive 650 miles?

6. Don sold 4 cars during the first week of January. If this rate stays

constant, about how many cars should he expect to sell in 1 year?

7. Carlie is able to walk ¾ mile in 30 minutes. Complete the graph to

determine how many minutes it would take Carlie to walk 1 mile.

8. A store sells apples in 5‐pound bags for $2.25. If the price per pound is

the same, how much will a 20‐pound bag of apples cost?

9. For every $25 spent at the candle store, the customer receives a

discount of $3. How much will a customer pay on a $75 purchase,

excluding tax?

Name: ____________________________________Date: ____________________

Evaluate: Rates

1. A 2‐pound can of coffee costs $6.50. Which of the following cans of coffee

would be the same price per pound?

A. A 1‐pound can of coffee for $4.50

B. A 3‐pound can of coffee for $7.50

C. A 5‐pound can of coffee for $11.50

D. A 7‐pound can of coffee for $22.75

2. Paige bought 3 bottles of orange juice for $4.50 and 2 bottles of apple juice

for $3.50. Based on these prices, which of the following statements is true?

A. The cost of a bottle of orange juice is $0.25 less than the cost of a bottle

of apple juice.

B. The cost of a bottle of apple juice is equal to the cost of a bottle of

orange juice.

C. The cost of a bottle of apple juice is one dollar less than the cost of a

bottle of orange juice.

D. The cost of 5 bottles of orange juice is $8.00

3. A local farmer’s market sells a 5‐pound bag of oranges for $2.75. Which of

the following bags of oranges would be the same price per pound?

A. A 3‐pound bag for $2.00

B. A 10‐pound bag for $7.75

C. A 15‐pound bag for $9.75

D. A 20‐pound bag for $11.00

4. Martha worked 16 hours last week. If she earned $152 last week, what was

her hourly rate?

A. $10.00

B. $9.50

C. $8.50

D. $6.00


Date post:	08-Mar-2018
Category:	Documents
Upload:	trinhthien
View:	214 times
Download:	2 times

Seventh Grade Tasks Weekly Enrichment Teacher · PDF file · 2014-07-03SOLVING MATH...

Documents