1. Solving Percent Problems with a Double Number Line Model

2. Warm Up
CONTENT OBJECTIVE: Students will be able to construct and interpret a double number line model for solving percent problems involving finding the whole or a part or the percent.
LANGUAGE OBJECTIVE: Students will be able to describe verbally to another student their understanding of double number lines. They will listen to the other student share.
1. The ratio of the number of girls to the number of boys in a chess club is 3 to 2. There are 14 boys in the chess club. What is the number of girls in the chess club?
2. A florist sells 8 roses for a total of $10. Each rose costs the same amount.
What is the cost of 12 roses? 21
girls
boys 7 7 7
(Time on this slide – 10 min) Time passed 10 min
In-Class Notes
This slide is intended to remind students of two ratio problem solving models they have already learned. They do not need to be prompted to use them. Just notice if students did use models and which ones. The answers are suggested methods of problem solving.
Preparation Notes
roses
cost 8
$10 4 $5 7 7
Answer:
21 girls
Answer:
$15 14 12
$15
Agenda

3. Launch Rihanna is a very popular musical artist.
Her 2012 album Unapologetic has sold close to 1 million copies worldwide.
The album can be bought at most places for $15. How much money would you expect that Rihanna has made from the sales of Unapologetic?
There are many costs that have to be paid from the sale of an album before the artist can receive her share of the profits.
Based on most information we can assume that Rihanna only received 5% of the total sales of the album. How much money is that?
(Time on this slide – 1 min) Time passed 12 min
In-Class Notes
Very quick slide to introduce a real-life context for problem solving using percents.
Preparation Notes
Agenda

4. Launch New Tool: Double Number Lines
Today we are going to learn how to calculate the percent of an amount.
We are going to use a tool called the Double Number Line.
To find out how much money Rihanna is estimated to have made from the sale of Unapologetic we need the total sales of the album and the percent she received.
$15 per album x 1 million albums sold = $15 million sales of all albums
Again, Rihanna only received about 5% of the total sales of the album.
(Time on this slide – 2 min) Time passed 14 min
In-Class Notes
This is only the launch. Do not let students get confused or wrapped up in the details. You may want them to focus on what is “fair” for an artist. If she sold 1 million albums and they are $15 a piece shouldn’t she get $15 million? Since it’s her music? Let’s see…
You can have students note that we are going to learn how to use a new tool: the double number line.
Preparation Notes
Let’s use a double number line to determine: How much money is that?
Agenda

5. Launch $0 $1.5 $15 percent Sales New Tool: Double Number Lines 50% 10%
$15 per album x 1 million albums sold = $15 million sales of all albums
How much money is 5% of $15 million?
Rihanna only received about 5% of the total sales of the album.
So Rihanna received about $750,000 from the sales
of an album that made $15,000,000.
$0.75 $0
$1.5
$15
percent
Sales
(in millions)
50%
10%
20%
30%
40%
60%
70%
80%
90% 0%
100% 5%
(Time on this slide – 2 min) Time passed 16 min
In-Class Notes
Do not spend much time describing how the tool works. The point is to let students see that there is a tool that can be used to find the percent of a number.
Preparation Notes
If this slide is the first time you have seen a double number line, do not stay on this slide too long. Preview the entire lesson first. The following slides detail step by step how to use a double number line.
$15 million sales worldwide
5% or $750,000
Agenda

6. Launch Let’s learn how to use the double number line tool.
New Tool: Double Number Lines
Let’s learn how to use the double number line tool.
(Time on this slide - min) Time passed
In-Class Notes
Preparation Notes
Agenda

7. Explore
New Tool: Double Number Lines
1.) Carly made 20 out of 80 shots while practicing basketball. What is the percentage of shots that Carly made? 80
percent
shots 0%
100%
We are going to make a double number line to compare
shots to percentage. Then we can easily find the percent.
She took 20 shots, so we have to put 20 in the right
position on the number line. First let’s find half.
First, we set up the zeroes. Zero shots is zero percent.
It says she took 80 total shots, so 80 is 100% of the shots.
(Time on this slide – 2 min) Time passed 17 min
In-Class Notes
Prompt students to follow along with the mini-lesson worksheet.
Preparation Notes
Agenda

8. Explore
1.) Carly made 20 out of 80 shots while practicing basketball. What is the percentage of shots that Carly made? 20 40
75% 60 80
percent
shots 0%
25%
50%
100%
Let’s fill out the other side of the number line.
Half of the 80 shots, is 40. And half of 100% is 50%.
The question asks for the percent that is equal to 20 out of 80.
So, where do I put the 20?
It’s half of 40!
(Time on this slide – 3 min) Time passed 20 min
In-Class Notes
Walk around and make sure that students are following along with the worksheet. It will help them to advance to the independent stage faster if they really dig in deep with the first few example problems.
Make sure students write the answer in the star shaped answer box on the worksheet.
Preparation Notes
You can see that every number of shots is lined up with a percent.
The answer is 25%.
We find halfway between 0 and 40. What percent is that?
Agenda

9. Explore
1.) Carly made 20 out of 80 shots while practicing basketball. What is the percentage of shots that Carly made?
25%
percent
shots 0%
100% 80
50% 40 20
75% 60
Partners: Take turns asking and answering:
(Time on this slide – 5 min) Time passed 25 min
In-Class Notes
“Lefty” and “Righty” are designations for the two people who make up a pair. Before beginning this activity make sure that students know who is on the left side of the pair (You are Lefty) and who is on the right side (Righty).
Students talk for 1-2 minutes depending on how productive they are with this practice.
You may have one Lefty and one Righty share out after they talk for one minute.
Preparation Notes
Lefty – Ask Righty “How did we figure out where to put 20?”
Righty – Ask Lefty “How do we know that 20 shots = 25%?”
Agenda

10. Explore – Let’s do the next one together!
2.) There are 6 candies in a bag that is 30% full. How many candies are in a full bag? ?
percent
candies 0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
This time we know the part of the bag and the percent but not the total. We can still use a double number line.
Next we need to put 30% in the right position. We can’t just put it anywhere. It has to be the right spot.
First we need to mark out zero and 100%
(Time on this slide – 3 min) Time passed 28 min
In-Class Notes
This problem is also on the mini-lesson class worksheet.
Take any ideas students may have about how to place 30%. You may wish to do another partner share if many people want to speak and have ideas. (Although there is more formal prompt to pair-share on the next slide.)
It is critically important that students make an effort to place the intervals on the number line equally spaced.
Preparation Notes
Many mistakes and misunderstandings can arise if the students set up the number line carelessly. Spaces between 0-10% and 10-20% (for example) must be the same size more or less. While spaces of different intervals must not appear to be the same distance (for example 0-10% must not equal 30-50%).
It doesn’t have to be perfect but many students make errors on number lines by not attempting to space the intervals accurately.
Do you have any ideas?
Agenda

11. Explore – Let’s do the next one together!
2.) There are 6 candies in a bag that is 30% full. How many candies are in a full bag? 6 ?
percent
candies 0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
We know that 30% is equal to 6 candies. So we can put 6 across from 30%.
(Time on this slide – 2 min) Time passed 30 min
In-Class Notes
Answers on the next slide.
Preparation Notes
Partner speaking is essential for the ELLs in the classroom and also helps general development of understanding for all learners. It also provides students with a change of pace to aid focus.
Can you use this information to find out how much 100% is?
Think, Pair, Share: See if you can find an answer. Then, tell your speaking partner. Listen to your partner’s idea.
Agenda

12. Explore – Let’s do the next one together!
2.) There are 6 candies in a bag that is 30% full. How many candies are in a full bag? 2 4 6 8 10 12 14 16 18 20
percent
candies 0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Here is one way to solve it. There are many ways.
Answer:
20 candies
(Time on this slide – 2 min) Time passed 32 min
In-Class Notes
The candies fill in automatically after you click with a short delay between numbers appearing. You will miss the auto-fill if you click too quickly on this slide.
Have students write the answer on the worksheet.
If time, you can have students share any thoughts they have about “counting by 2’s” versus dividing or multiplying.
Preparation Notes
As students’ skills improve they may begin to notice that 100% divided by 10 is 10% so the number on the other side of 100% can be divided by 10 to give the 10% value and then multiplied (by say 3) to get other percents (say 30%).
Agenda

13. On Your Own – Try this one!
3.) A class conducts a survey of 1,000 students. The survey reveals that 20% of the students speak Spanish. How many students is this?
percent
students 0%
100%
1,000
100
10%
200
20%
300
30%
400
40%
500
600
60%
700
70%
800
80%
900
90%
50%
(Time on this slide – 3 min) Time passed 35 min
In-Class Notes
Reveal the process step by step as needed. Each piece of the animation can be used as a hint for solving the problem.
Take your time clicking through the animation. The interval lines on the number line and the numbers for each interval will slowly fill in if you give the animation time to continue.
This may more efficiently be solved by splitting the 100% into five parts (20%, 40%, 60%, 80% and 100%). Feel free to show this solution as well.
Preparation Notes
Answer: 200 students speak Spanish.
Agenda

14. Percents in Life taxes discounts tips
Here are a few examples of situations in which adults calculate
with percents in real life.
(Time on this slide – 1 min) Time passed 36 min
In-Class Notes
Preparation Notes
taxes
discounts
tips
Agenda

15. Taxes and Tips are a percent added on to a cost.
4.) Cheryl had breakfast in the diner and the bill came to $ She
would like to leave a 20% tip. How much should she leave altogether?
5.) Martin wants to purchase a book that costs $12. The sales tax
Is 5%. If Martin has $13, can he purchase the book?
Both of these problems can be solved with the same double number line.
percent
dollars 0%
100%
$12
(Time on this slide – 1 min) Time passed 37 min
In-Class Notes
Preparation Notes
Agenda

16. Taxes and Tips are a percent added on to a cost.
4.) Cheryl had breakfast in the diner and the bill came to $ She
would like to leave a 20% tip. How much should she leave altogether?
Just remember to add the tip to the total amount if you are looking
for the total cost!
Answer: $ $2.40 = $14.40
percent
dollars 0%
100%
$12.00
$1.20
10%
$2.40
20%
$3.60
30%
$4.80
40%
$6.00
$7.20
60%
$8.40
70%
$9.60
80%
$10.80
90%
50%
(Time on this slide – 3 min) Time passed 40 min
In-Class Notes
Slowly click through the animation to allow the intervals and amounts to fill in.
Can also be solved using fifths.
Prompt students to write what to remember in the box on the worksheet.
This is also a great opportunity for a “partner speak” as time permits (“What do you need to remember about problems involving a tip?”
Preparation Notes
You may wish to prompt students to think about how we would know that 10% is $1.20.
One possible idea: I know that half of $12 is $6 and then I know there are five jumps from 0 – 50% so $6 divided by 5 is $1.20.
Also: I want students to begin to notice that 10% is 100% divided by 10. $12 is $1.20 (either through long division or even better by knowing how to move the decimal point for division by 10).
Agenda

17. Taxes and Tips are a percent added on to a cost.
5.) Martin wants to purchase a book that costs $12. The sales tax
Is 5%. If Martin has $13, can he purchase the book?
How do you find 5% on this number line? Do you have any ideas?
5% is halfway between 0% and 10%. What is half of $1.20?
percent
dollars 0%
100%
$12.00
$1.20
10%
$2.40
20%
$3.60
30%
$4.80
40%
$6.00
$7.20
60%
$8.40
70%
$9.60
80%
$10.80
90%
50%
(Time on this slide – 3 min) Time passed 43 min
In-Class Notes
This can be a little hard to draw.
Maybe some students remember from the Launch example?
Preparation Notes
Agenda

18. Taxes and Tips are a percent added on to a cost.
5.) Martin wants to purchase a book that costs $12. The sales tax
Is 5%. If Martin has $13, can he purchase the book?
Don’t forget that a tax is also an amount added on to the total.
Answer: He does have enough money because $ $0.60 = $12.60 total cost with tax.
$0.60
percent
dollars 0%
100%
$12.00
$1.20
10%
$2.40
20%
$3.60
30%
$4.80
40%
$6.00
$7.20
60%
$8.40
70%
$9.60
80%
$10.80
90%
50%
(Time on this slide – 2 min) Time passed 45 min
In-Class Notes
Prompt students to use the worksheet.
See notes on previous slide regarding the animation and prompting students to think about values on the number line.
Preparation Notes 5%
Agenda

19. Discounts are an amount taken OFF of a total cost.
6.) Sam got a $12 discount off a $48 purchase. What percent
discount did he get?
The question could have asked: how much did Sam pay for a purchase that was 25% off of the original price of $48. What would the answer be?
I can see that $24 is half of $48. And $12 is half of $24.
So, what’s half of 50%?
$ $12.00 = $36.00 is the cost of the purchase after a 25% discount.
percent
dollars 0%
100%
$48
$12
$24
Answer:
25%
50%
(Time on this slide – 3 min) Time passed 48 min
In-Class Notes
Prompt students to use the worksheet
Preparation Notes
25%
Remember: If the question asks for the total cost you must subtract the discount.
Agenda

20. Try These! Independent Practice
1.) Alexis bought a CD player. She does not remember the price, but
she does know that the 5% sales tax came to $ What was the price of the CD player?
2.) Customers left Jill $2.50 as a tip. The tip was 20% of the total. How
much was the bill?
(Time on this slide - min) Time passed
In-Class Notes
These problems are found on the independent practice worksheet. They can be used as part of the same class if you have a longer block schedule or a separate class period.
Preparation Notes
Agenda

21. What is the answer to this problem?
Summary
1.) Carly made 20 out of 80 shots while practicing basketball. What is the percentage of shots that Carly made?
25%
percent
shots 0%
100% 80
50% 40 20
75% 60
What is the answer to this problem?
Partners – Take turns describing how to solve this problem, step by step.
(Time on this slide – 4 min) Time passed 52 min
In-Class Notes
The summary may also be written in a notebook or on a sheet of paper to be turned in.
Preparation Notes
Agenda