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LCM, GCF and Operations with Fractions and DecimalsEnrichment Investigation #1

Reality in A Restaurant

Common Core State Standard(s):6.NS.1 Dividing Fractions6.NS.2 Divide Multi-Digit Numbers

Standard(s) for Mathematical Practice: Make sense of problems and persevere in

solving them Reason abstractly and quantitatively Attend to Precision

Materials Needed: Reality in a Restaurant Student Task Sheet Calculators Technology Access if needed to create presentation and/or visuals

Instructions to Teacher: Students are given a problem-based simulation dealing with ordering food for a Pizza Franchise. Students must read through the problem and the task and create a presentation that includes all of the criteria in the task. This presentation can be in any style they wish, but must include detailed representations of the mathematics used. It is the teacher’s discretion as to whether calculators can be used or not. Students will need a copy of the Student Task Sheet, “Reality in a Restaurant” and access to technology resources if they are available and the student would like to create their presentation with technology. An answer key is provided that answers the essential questions posed in the problem. However, in the presentation, students need to include charts and reflections as asked for in the task. A rubric for assessment is included. A copy of the student task is below.

Reality in a Restaurant Investigation for Students:Student Problem: You have the opportunity to buy a local Pizza and Salad Franchise in your community. As you consider the purchase, you want to be sure this is something you really want to do. The manager of the restaurant explains the tedious work in ordering supplies from week to week and how it is imperative to keep track of the food used in order to order efficiently for as much profit as possible as well as the least amount of food wastage. The manager tells you that business is very good and the restaurant has more customers than ever! You are going to do a little investigating to find the truth of the matter.

Task: Running account of last week’s lettuce use is in the table below. You need to get an idea of how much lettuce you will be ordering week to week for customer salads. Customers can order a salad made from iceburg lettuce or a salad made from Romain lettuce. From this you will also investigate how many salads were prepared in a week and the profit you made on the salads.

Wake County Public Schools, 2012

INVESTIGATIONS TO ANSWER: ORGANIZE YOUR INFORMATION CAREFULLY AND THOUGHTFULLY. INCLUDE CHARTS FOR VISUALS AND SHOW ALL WORK NEATLY. PREPARE A PRESENTATION OF YOUR FINDINGS. INCLUDE REFLECTIONS ON WHAT YOUR NEW UNDERSTANDINGS ARE ABOUT THIS RESTAURANT AND ITS PROFITS FROM SALADS. DON’T FORGET THAT THERE MAY BE SOME OTHER COSTS IN THE SALAD AS THIS JUST INCLUDES LETTUCE, ONION AND TOMATO. WHAT ELSE MIGHT BE INCLUDED? INCLUDE ANY UNANSWERED QUESTIONS OR CONSIDERATIONS THAT WOULD BE GOOD TO THINK ABOUT AND WHY.

1. The amount of Iceburg and Romain Lettuce to order for the next week along with pounds of onions and tomatoes

2. Cost of a week’s lettuce order3. Total customer salads made in a week4. How many pounds of onions and tomatoes were needed for just salads in a week5. Cost of a week’s onions and tomatoes6. Total profits from customer salads7. Reflections – include new understandings and things you may change in the order

Useful Information: A head of Iceburg lettuce ordered per week costs on average $1.39 per head A head of Romaine lettuce ordered per week costs on average $1.60 per head A customer salad uses approximately 1/5 head iceburg lettuce (Romaine lettuce is mixed in) A customer salad uses 1/6 of a large onion and 1/3 of a large tomato The restaurant charges $2.50 for a salad One large red onion weighs in at about 13. 3 ounces and the cost per pound is $1.21 One large red tomato weighs in at about 6 ounces and the cost per pound is $1.99 There are 16 ounces in a pound

Table of Lettuce used this past week:Sunday 2 ¾ Iceburg Lettuce 1 ½ Romain LettuceMonday 1¾ Iceburg Lettuce ½ Romain LettuceTuesday 1 ¼ Iceburg Lettuce ¼ Romain LettuceWednesday 1 ½ Iceburg Lettuce 1 Romain LettuceThursday 1 ½ Iceburg Lettuce ½ Romain LettuceFriday 3 ½ Iceburg Lettuce 2 Romain LettuceSaturday 3 ¼ Iceburg Lettuce 1 ¼ Romain Lettuce

Sources: Real Life Math Mysteries by Mary Ford Washington Hittoon.com (Clip art)


Investigation: Reality in a Restaurant Generalization: Success in entrepreneurship involves mathematical problem solving.

Problem: You have the opportunity to buy a local Pizza and Salad Franchise in your community. As you consider the purchase, you want to be sure this is something you really want to do. The manager of the restaurant explains the tedious work in ordering supplies from week to week and how it is imperative to keep track of the food used in order to order efficiently for as much profit as possible as well as the least amount of food wastage. The manager tells you that business is very good and the restaurant has more customers than ever! You are going to do a little investigating to find the truth of the matter.

Task: Running account of last week’s lettuce use is in the table below. You need to get an idea of how much lettuce you will be ordering week to week for customer salads. Customers can order a salad made from iceburg lettuce or a salad made from Romain lettuce. From this you will also investigate how many salads were prepared in a week and the profit you made on the salads.

INVESTIGATIONS TO ANSWER: ORGANIZE YOUR INFORMATION CAREFULLY AND THOUGHTFULLY. INCLUDE CHARTS FOR VISUALS AND SHOW ALL WORK NEATLY. PREPARE A PRESENTATION OF YOUR FINDINGS. INCLUDE REFLECTIONS ON WHAT YOUR NEW UNDERSTANDINGS ARE ABOUT THIS RESTAURANT AND ITS PROFITS FROM SALADS. DON’T FORGET THAT THERE MAY BE SOME OTHER COSTS IN THE SALAD AS THIS JUST INCLUDES LETTUCE, ONION AND TOMATO. WHAT ELSE MIGHT BE INCLUDED? INCLUDE ANY UNANSWERED QUESTIONS OR CONSIDERATIONS THAT WOULD BE GOOD TO THINK ABOUT AND WHY.

8. The amount of Iceburg and Romain Lettuce to order for the next week along with pounds of onions and tomatoes9. Cost of a week’s lettuce order10. Total customer salads made in a week11. How many pounds of onions and tomatoes were needed for just salads in a week12. Cost of a week’s onions and tomatoes13. Total profits from customer salads14. Reflections – include new understandings and things you may change in the order

Useful Information:

A head of Iceburg lettuce ordered per week costs on average $1.39 per head A head of Romaine lettuce ordered per week costs on average $1.60 per head A customer salad uses approximately 1/5 head iceburg lettuce (Romaine lettuce is mixed in) A customer salad uses 1/6 of a large onion and 1/3 of a large tomato The restaurant charges $2.50 for a salad One large red onion weighs in at about 13. 3 ounces and the cost per pound is $1.21 One large red tomato weighs in at about 6 ounces and the cost per pound is $1.99 There are 16 ounces in a pound

Table of Lettuce used this past week:

Sunday 2 ¾ Iceburg Lettuce 1 ½ Romain LettuceMonday 1¾ Iceburg Lettuce ½ Romain LettuceTuesday 1 ¼ Iceburg Lettuce ¼ Romain LettuceWednesday 1 ½ Iceburg Lettuce 1 Romain LettuceThursday 1 ½ Iceburg Lettuce ½ Romain LettuceFriday 3 ½ Iceburg Lettuce 2 Romain LettuceSaturday 3 ¼ Iceburg Lettuce 1 ¼ Romain Lettuce6th Grade Math Performance Rubric for Enrichment Investigations


Student Name: ___________________________________________

Categories to Be Scored

Insufficient=1

18 points

Progressing=2

21 points

Proficient=3

23 points

Exemplary=4

25 points

Time Management and Task Commitment

Score: _____

Investigation incomplete.

Needed frequent assistance. Effort is incomplete or not adequate for the investigation.

Used time appropriately and effort was successful.

Managed time in a mature manner making independent decisions.

Extensive commitment. Rigorous effort in responsible manner.

Organization of Investigation

Score: _____

Unclear, lacks organization

Attempts made to organize and sequence but investigation product is hard to follow.

Organized effectively. Clear sequence.

Skillfully planned with logical sequence. Very clear and communicated well.

Complexity in Explanations of Reasoning

Score: ______

Too simple or not appropriate.

Simple information presented in explanation. Lack of details shown. Work to show deeper understanding, connection and application of ideas.

Shows complexity by communicating deeper understanding through details and thinking skills. Reasonable connections are made that are clear and logical.

Shows complexity by communicating deep understanding through detailed explanations and thinking skills beyond expected level. May show multiple points of view, compare/contrast and make generalizations. Clear, logical and possibly creative connections are made.

Problem Solving

Inappropriate with lack of

Incomplete or partially inaccurate

Appropriate problem solving

High level solutions showing deep


Approaches and Strategies

Score: _____

accurate problem solving approaches used.

use of problem solving strategies.

strategies used with accuracy and effective analysis shown. Flexibility with strategies is shown.

understanding of problem solving strategies and flexibility with strategies. High accuracy shown. Synthesizes and evaluates in the process.

Total Score: ________

Comments:

Answer Key: Reality in a Restaurant

The amount of Iceburg and Romain Lettuce to order for the next week = 16 head iceburg and 7 Romaine Add all the Iceburg in the chart and you get 15 ½ heads and add all Romaine and you get exactly 7.

Price Iceburg = $1.39 x 16 = $22.24 Price Romaine =$1.60 x 7 = $11.20 Total = $33.44

Total customer salads made in a week = 15 ½ ÷ 1/5 = 77 ½ or approximately 78 salads

How many pounds of onions and tomatoes were needed for just salads in a week:

Onions = 78 salads x 1/6 onion per salad = 13 onions / 13.3oz per onion x 13 onions = 172.9 oz ÷ 16 = 10.8lbs or approximately 11 lbs onions

Tomatoes = 78 salads x 1/3 tomato per salad = 26 tomatoes / 6 oz per tomato x 26 tomatoes = 156 oz 156oz ÷ 16 = 9.75 lb or approximately 10 lbs tomatoes

Cost of tomatoes and onions :

Onion cost = 11 lbs x $1.21/lb = $13.31 Tomato cost = 10 lbs x $1.99/lb = $19.90 Total cost = $33.21

Total profits from customer salads:


Cost of lettuce ($33.44) + cost of onions and tomatoes ($33.21) = $66.65 78 customer salads @ $2.50/salad = 78 x $2.50 = $195.00 Profit = 195 – 66.65 = $128.35

Useful Information:

A head of Iceburg lettuce ordered per week costs on average $1.39 per head A head of Romaine lettuce ordered per week costs on average $1.60 per head A customer salad uses approximately 1/5 head iceburg lettuce and 1/8 head of Romain lettuce A customer salad uses 1/5 onion and 1/3 of a tomato The restaurant charges $2.50 for a salad One large red onion weighs in at about 13. 3 ounces and the cost per pound is $1.21 One large red tomato weighs in at about 6 ounces and the cost per pound is $1.99 There are 16 ounces in a pound

Table of Lettuce used this past week:

Sunday 2 ¾ Iceburg Lettuce 1 ½ Romain LettuceMonday 1¾ Iceburg Lettuce ½ Romain LettuceTuesday 1 ¼ Iceburg Lettuce ¼ Romain LettuceWednesday 1 ½ Iceburg Lettuce 1 Romain LettuceThursday 1 ½ Iceburg Lettuce ½ Romain LettuceFriday 3 ½ Iceburg Lettuce 2 Romain LettuceSaturday 3 ¼ Iceburg Lettuce 1 ¼ Romain Lettuce

LCM, GCF and Operations with Fractions and Decimals

Enrichment Investigation #2The Scuba Scenarios

Common Core State Standard(s):6.NS.3 Add, subtract, multiply and divide multi-digit decimals


solving them Reason abstractly and quantitatively Attend to Precision

Materials Needed: Technology Access to watch video Scuba Scenario Task for Students Article to read – “How Deep Can you Scuba?” Materials of student choice to create brochure. This can be technology based.

Instructions to Teacher: Students will gather some background information on scuba diving and the mathematics involved in the sport. They work with a problem-based simulation culminating with a brochure or presentation for a product. There is a video that students can watch that describes the equipment necessary for scuba diving and an article to read.


Students should be given the “Scuba Scenario” Investigation (this is below, but students get a separate copy). They should read the background information and follow the criteria/questions involved to complete the brochure. A rubric for assessment is included.

Investigation for Students: Scuba Scenario Generalization: Recreation in Life consists of an understanding of basic mathematical concepts.

Problem: You have been hired by a Scuba Diving Touring Company in Mexico to create an informational brochure for their beginner class. You have never actually been diving, but your mathematical skills are needed for the brochure. You will need to learn the basics of scuba diving and its risks to properly design your brochure. Mathematics is essential to decrease some of the risks, especially as the math relates to underwater air supply and how depth affects the amount of time that a person can spend underwater. After analyzing the background information below, looking at the provided resources and calculating answers to the questions, you must include all of this in a brochure or brochure-type presentation to display the information that you have learned in order to aid the beginner class for the Touring Company.

Background Information:The word “SCUBA” stands for “self contained underwater breathing apparatus. Just as you take in air on land, you must also take in air when you are underwater. However, you must artificially provide yourself that air. The amount of time you can spend with your air underwater depends upon the depth you dive. The deeper you go, the more air you will require.

Explanation of Air Pressure under the Water (Excerpt from Chapter 24, Real Life Math Investigations, by Edward Zaccaro)“On land, at sea level, there are about 15 pounds of pressure for every square inch of space. 15 pounds per square inch, or (psi) is an amount of air pressure that is called one atmosphere. Right now, you have one atmosphere of air pressure pushing against your body. In fact, on each side of your hand right now, there are about 250 pounds of air pushing against your skin! And that is just one of your hands! Just think of how much air is pushing against the rest of your body! The reason we don’t feel this pressure is because our bodies are mostly made of water, and liquids cannot be squeezed.

Liquids cannot be squeezed, or compressed, but gases like air can. If you take an empty glass, turn it upside down and put it all the way under water, the air stays in the glass, and no water gets in. What do you think will happen to the air in that glass when you bring it down to 33 feet? The air will be squeezed into a smaller space because of all the water pressing on it from above. At 33 feet, the air will be squeezed into a space half as big as it was at the surface.

Remember when we talked about air pressure at sea level? Above water, you have one atmosphere of pressure pushing against you. One atmosphere, you remember, is equal to about 15 pounds per square inch. While under water, you only need to go to a depth of 33 feet to double that pressure. So at this depth, we have two atmospheres pushing against us, or 30psi. For every 33 feet of depth, we add another atmosphere of pressure. What will happen if we go down to 66 feet and 3 atmospheres of pressure?....the glass is 1/3 full of air. The amount of air in the glass can be expressed as a fraction of the whole. In this case, 1/3. The amount of space the air takes up as a fraction of the whole is the reciprocal of how many atmospheres of pressure are pushing on it. So if we were to go even further down, to 99 feet, or 4 atmospheres of pressure, the glass would be only ¼ full of air, and that air would be four times as dense!”


More Background Information: Explanation of Air Tank, excerpt taken from Ch 24, Real Life Math Investigations, by Edward Zaccaro “We need to talk about the air tank that comes underwater with you. It is about 2 feet long and about 7-8 inches wide. When filled, this tank holds about as much air as fits in a telephone booth. All of that air is compressed down to that small size to fit on your back! Inside a filled scuba tank, there are about 3000 pounds of pressure per square inch, or 3000 psi!

Now let’s take an empty balloon with us, instead of a glass. We’ll go down to 33 feet and fill our balloon with air. What do you think will happen to this balloon if we take it to the surface? Will it get smaller or larger?

There were 2 atmospheres of pressure at 33 feet, now there is only 1, so the balloon got twice as big. If we filled our empty balloon at 66 feet and brought it to the surface, it would get 3 times as big, or it would pop before we got there! When you breathe at the surface, you are taking in about a liter of air each time you inhale. If you are underwater with your scuba tank at 33 feet, you are still taking in about a liter of air, but since you are at 30 psi of pressure, or 2 atmospheres, you are taking in twice the amount of air molecules with each breath. The air is twice as dense, therefore, you are pulling air out of your tank twice as fast as you would at the surface. It is now clear that the deeper you go, the less time you have before your air runs out. “

Basic Criteria for Brochure or Presentation: (Formats- Powerpoint/Prezi/webpage/publisher/hand crafted)

Interesting and catchy cover/Title Why Mathematics is essential for Scuba Diving Brief explanation or visual graphic organizer of information above. Include a visual that shows

and explains how many atmospheres of pressure a filled scuba tank has inside. Include an explanation of how deep you would have to go to have the pressure inside a filled scuba tank be equal to the pressure of the surrounding water.

A Chart to show the comparisons of depth, psi, atmospheres and fullness of air in a glass. Read the article, How Deep Can You Scuba Dive? Create your chart to go up to the depth of a recreational diver. You can go deeper if you like. You should include reflections on the patterns that are seen in your chart. With a visual describe what fraction of a glass will be filled with air if you take it upside down to a depth of 49.5 feet.

Balloon Scenarios –1. Describe a balloon with 3 liters of volume at the surface that is taken underwater until its

volume decreased to 1.5 liters and explain how deep it has to be.2. Describe taking an empty balloon (that can only hold 4 liters of air maximum) to a depth

of 132 feet and filling it with one liter of air. Explain whether the balloon will survive the trip back up to the surface or not with visuals.

Big Bob and Little Larry Scenario Cartoon/Comic/Visual1. In this scenario, compare how fast air will be used up when two people are diving. Big

Bob, a large person who breathes at the rate of 1.5 liters of air per breath, is swimming at a depth of 33 feet. He breathes every 5 seconds. Little Larry, a small person, breathes in


0.8 liters per breath and breathes once every 4 seconds.

Sources: 25 Real Life Math Investigations by Edward Zaccaro http://youtu.be/Ie3lpStLQ5Q Scuba Diving Equipment Requirements http://youtu.be/Ie3lpStLQ5Q First Scuba Diving Lessons Credits: clipartguide.com (Image of Man in Scuba Suit) About.com – Article- How Deep can you Scuba Dive?

Article for Students: How Deep Can You Scuba Dive?By Natalie Gibb, About.com Guide

“How deep have you been underwater?” One of my open water course students asked. This is a tricky question, one that I don't like to answer because I fear that my students may aspire to my maximum depth, or worse, attempt to beat it. A more appropriate question is, “How deep can scuba divers descend?” Unfortunately, the answer is not straightforward – it depends on a variety of factors such as breathing gas, experience level, and personal tolerance for high partial pressures of inert gasses and oxygen

What Is the Deepest a Scuba Diver Has Descended?:

The current depth record for open circuit scuba diving is held by Pascale Bernabé, who descended to 1082 feet (330 meters) in 2005. Nuno Gomez is the Guinness World Record holder for the deepest open circuit scuba dive, with a confirmed dive to 1043 feet (318 meters), also in 2005.

More Importantly, How Deep Can YOU Dive?:

Most recreational scuba diving organizations set the maximum depth for a certified, experienced recreational divers breathing air at 130 feet. Divers should heed this guideline. The fact that extremely experienced, technical divers have descended beyond 1000 feet on admittedly risky dives does not mean that recreational divers have any business breaking suggested depth limits. When a diver considers the reasons behind established depth limits, it becomes obvious why breaking depth guidelines is foolish.

Considerations in Determining a Maximum Depth:

• Decompression StatusThe deeper a diver descends, the shorter his no-decompression limit will be. For example, a diver who descends to 40 feet can remain at the depth for 140 minutes (air supply permitting). A diver who descends to 130 feet can stay only 10 minutes at that depth before accumulating so much nitrogen in his body that he requires a series of decompression stops to reduce his risk of decompression sickness. Descending beyond 130 feet without decompression dive training does not allow a diver much time to enjoy his dive.

• Air ConsumptionA diver breathes air at the pressure of the water around him (ambient pressure). The deeper a diver goes, the more the air


http://scuba.about.com/od/certificationopenwater/a/openwatercert.htm

http://youtu.be/Ie3lpStLQ5Q

http://youtu.be/Ie3lpStLQ5Q

he breathes compresses (learn more about water pressure and diving). At a depth of 130 feet, a diver consumes his air approximately five times faster than he does on the surface. Divers who plan on diving to this depth will find that their dive time is limited by air consumption. Not only will a diver use his air more quickly at greater depths, he will require a large air reserve for the long ascent from deep dives.

• NarcosisSome gases, such a nitrogen, may cause narcosis in divers at increased partial pressures. Every diver will experience this narcosis eventually, but the onset of inert gas narcosis varies from diver to diver and from day to day. Be warned - even if you experience the drunken feeling of narcosis as enjoyable, it shares many of the symptoms of alcohol intoxication such as impaired motor coordination, judgement, and reasoning. Some divers even report visual disturbances and a skewed sense of time. This is not a good state to be in when deep underwater. A diver should slowly increase dive depths as he gains experience and he should be sure to make his initial deep dives (deeper than 60 ft) with a qualified individual, such as a guide or instructor, who can monitor him for signs of narcosis and assist him if necessary.

• Oxygen ToxicityAt very high concentrations, oxygen becomes poisonous (oxygen toxicity), causing convulsions, unconsciousness, and even death. When the recreational depth guidelines are followed, oxygen toxicity is not a concern for scuba divers. Still, this give divers another (very good) reason not to exceed depth limitations. The oxygen in air may become toxic at depths beginning at approximately 218 feet, and gas mixtures with high percentages of oxygen, such as enriched air nitrox, maybe be toxic at much shallower depths.

• Experience LevelDepth is a stress factor in scuba diving. Psychologically, deeper dives are stressful because divers are farther from their exit point. Divers will notice their air supply dropping more rapidly than at shallower depths, may notice an increase in breathing resistance, and are likely to experience some form of mild narcosis. While deeper dives are frequently very beautiful, have pristine reefs, and different wildlife than shallow dives, divers should increase their dive depths cautiously. Making your first deep dives under the supervision of a qualified guide or instructor is always advisable.

What Are Common Depth Limits for Recreational Certification Levels?:

The suggested depth guidelines for various recreational scuba diving certifications vary among organizations. In general:

Adults • Experience Courses (e.g. PADI's Discover Scuba Diving) - 40 feet/12 meters• Subsequent Dives for Non-Certified Divers - 40 feet/12 meters• First and Second Training Dives - 40 feet/ 12 meters• Dives 3 and 4 of Open Water Training - 60 feet/ 18 meters• Open Water Certified Divers - 60 feet/ 18 meters• Experienced Certified Divers, or Divers With Advanced/ Deep Training - 130 feet/ 40 meters Children • Children Ages 8 - 9 (First Dive) - 6 feet/ 2 meters• Children Ages 8 - 9 (Successive Training Dives) - 12 feet/ 4 meters• Children Ages 10 -11 (Open Water Certified) - 40 feet/ 12 meters Teens • Teenagers Ages 12 - 14 (Open Water Certified) - 60 ft/ 18 meters• Teenagers Ages 12 - 14 (Advanced Certifications) - 70 feet/ 21 meters• Teenagers Ages 15 and Over - Same as adult limitations

How Can a Diver Safely Exceed These Depth Limits?:

Divers can descend deeper than 130 feet. In fact, they do it all the time. However, diving deeper than 130 feet requires technical dive training, such as deep air, decompression procedures, and trimix courses. Never attempt to dive deeper than the recreational dive limits without specialized training.


6th Grade Math Performance Rubric for Enrichment Investigations

Student Name: ___________________________________________


Insufficient=1

18 points

Progressing=2

21 points

Proficient=3

23 points

Exemplary=4

25 points


Score: _____







Score: _____







Score: ______





Problem Solving Approaches and Strategies

Score: _____

Inappropriate with lack of accurate problem solving approaches used.

Incomplete or partially inaccurate use of problem solving strategies.

Appropriate problem solving strategies used with accuracy and effective analysis shown. Flexibility with strategies is shown.

High level solutions showing deep understanding of problem solving strategies and flexibility with strategies. High accuracy shown. Synthesizes and evaluates in the process.


Comments:


Scuba Scenario Answers to be Included in Final Product

Basic Criteria for Brochure or Presentation: (Formats- Powerpoint/Prezi/webpage/publisher)

Interesting and catchy cover/Title – This will be open ended for students to show creativity Why Mathematics is essential for Scuba Diving- Look for critical explanations Brief explanation or visual graphic organizer of information above. Include a visual that shows

and explains how many atmospheres of pressure a filled scuba tank has inside. Include an explanation of how deep you would have to go to have the pressure inside a filled scuba tank be equal to the pressure of the surrounding water. – A filled scuba tank has 3000 pounds of pressure per square inch, or 3000psi. One atmosphere is 15psi so 3000÷15 = 200 atmospheres. To have the pressure inside the tank be equal to the pressure of the surrounding water you would have to go down 6,567 feet (more than a mile underwater). Students may think you multiply 200 by 33 but you really multiply 199 by 33 because 33 feet down is TWO atmospheres and not one. Therefore, you have to subtract one atmosphere before multiplying.

A Chart to show the comparisons of depth, psi, atmospheres and fullness of air in a glass. Read the article, How Deep Can You Scuba Dive? Create your chart to go up to the depth of a recreational diver. ( According to the article that students should read, a recreational diver usually goes no deeper than 130 feet.) You can go deeper if you like. You should include reflections on the patterns that are seen in your chart. With a visual describe what fraction of a glass will be filled with air if you take it upside down to a depth of 49.5 feet. Sample Chart below. Students should describe the patterns they see. If you take the glass down to a depth of 49.5 feet, 2/5 of the glass will be full of air. Students should have noticed that pattern that the amount of air space is a reciprocal of how many atmospheres of pressure are pushing on it. At 49.5 feet, there are 33 + half of 33 (16.5) which is 2 ½ atmospheres or 5/2 atmospheres. Therefore the glass will be 2/5 full of air.

Depth Psi Atmospheres Fullness of air in glassSurface of water 15 1 Full33ft 30 2 ½ full66ft 45 3 1/3 full99ft 60 4 ¼ full132 75 5 1/5 full


Balloon Scenarios –3. Describe a balloon with 3 liters of volume at the surface that is taken underwater until its

volume decreased to 1.5 liters and explain how deep it has to be. The balloon has to be 33 feet deep. The volume of air at the surface decreases by half at a depth where there are two atmospheres of pressure. Since the volume of the balloon decreased by exactly ½, that is like the cup being ½ full of air and that would be at two atmospheres or 33 feet.

4. Describe taking an empty balloon (that can only hold 4 liters of air maximum) to a depth of 132 feet and filling it with one liter of air. Explain whether the balloon will survive the trip back up to the surface or not with visuals. Students should be creative in their visuals. The balloon will actually NOT survive the trip back up to the surface – it would pop. You take it down 132 feet, which is 5 atmospheres according to the chart. So, upon returning to the surface the balloon would swell to 5 liters (remember that at 5 atmospheres the balloon will hold 1/5 the air as at the surface.) So, since the balloon can only hold 4 liters, 5 liters would be too much!

Big Bob and Little Larry Scenario Cartoon/Comic/Visual2. In this scenario, compare how fast air will be used up when two people are diving. Big

Bob, a large person who breathes at the rate of 1.5 liters of air per breath, is swimming at a depth of 33 feet. He breathes every 5 seconds. Little Larry, a small person who is swimming at a depth of 49.5 feet, breathes in 0.8 liters per breath and breathes once every 4 seconds. The answer is that Bob, the large person, is using more air at 36 surface liters per minute while Little Larry is using 30 surface liters per minute.

Explanation – Big Bob, at 2 atmospheres (33 ft), is using the equivalent of 3 surface liters per breath. This is calculated by 1.5 x 2 atmospheres = 3. Since he breathes every 5 seconds, he is breathing 12 times per minute (60 seconds ÷ 5 = 12). At 3 surface liters per breath, Big Bob is using 36 surface liters of air per minute. (3 surface liters per breath x 12 breaths per minute).

Little Larry, is at 2.5 atmospheres because 33 feet is two atmospheres and if you add 16.5 (half of 33) you are adding on another half of an atmosphere to get to 49.5 feet. At 2.5 atmospheres, Larry is using the equivalent of 2 surface liters per breath (0.8 x 2.5 atmospheres). Since he breathes every 4 seconds, he is breathing 15 times per minute ( 60 ÷4 = 15). At 2 surface liters per breath, Larry is using 30 surface liters of air per minute. (2 surface liters per breath x 15 breaths per minute.)


Student Investigation: Scuba Scenario Generalization: Recreation in Life consists of an understanding of basic mathematical concepts.

Problem: You have been hired by a Scuba Diving Touring Company in Mexico to create an informational brochure for their beginner class. You have never actually been diving, but your mathematical skills are needed for the brochure. You will need to learn the basics of scuba diving and its risks to properly design your brochure. Mathematics is essential to decrease some of the risks, especially as the math relates to underwater air supply and how depth affects the amount of time that a person can spend underwater. After analyzing the background information below, looking at the provided resources and calculating answers to the questions, you must include all of this in a brochure or brochure-type presentation to display the information that you have learned in order to aid the beginner class for the Touring Company.

Background Information:

The word “SCUBA” stands for “self contained underwater breathing apparatus. Just as you take in air on land, you must also take in air when you are underwater. However, you must artificially provide yourself that air. The amount of time you can spend with your air underwater depends upon the depth you dive. The deeper you go, the more air you will require.

Explanation of Air Pressure under the Water (Excerpt from Chapter 24, Real Life Math Investigations, by Edward Zaccaro)

“On land, at sea level, there are about 15 pounds of pressure for every square inch of space. 15 pounds per square inch, or (psi) is an amount of air pressure that is called one atmosphere. Right now, you have one atmosphere of air pressure pushing against your body. In fact, on each side of your hand right now, there are about 250 pounds of air pushing against your skin! And that is just one of your hands! Just think of how much air is pushing against the rest of your body! The reason we don’t feel this pressure is because our bodies are mostly made of water, and liquids cannot be squeezed.

Liquids cannot be squeezed, or compressed, but gases like air can. If you take an empty glass, turn it upside down and put it all the way under water, the air stays in the glass, and no water gets in. What do you think will happen to the air in that glass when you bring it down to 33 feet? The air will be squeezed into a smaller space because of all the water pressing on it from above. At 33 feet, the air will be squeezed into a space half as big as it was at the surface.

Remember when we talked about air pressure at sea level? Above water, you have one atmosphere of pressure pushing against you. One atmosphere, you remember, is equal to about 15 pounds per square inch. While under water, you only need to go to a depth of 33 feet to double that pressure. So at this depth, we have two atmospheres pushing against us, or 30psi. For every 33 feet of depth, we add another atmosphere of pressure.

What will happen if we go down to 66 feet and 3 atmospheres of pressure?....the glass is 1/3 full of air. The amount of air in the glass can be expressed as a fraction of the whole. In this case, 1/3. The amount of space the air takes up as a fraction of the whole is the reciprocal of how many atmospheres of pressure are pushing on it. So if we were to go even further down, to 99 feet, or 4 atmospheres of pressure, the glass would be only ¼ full of air, and that air would be four times as dense!”


More Background Information: Explanation of Air Tank, excerpt taken from Ch 24, Real Life Math Investigations, by Edward Zaccaro

“We need to talk about the air tank that comes underwater with you. It is about 2 feet long and about 7-8 inches wide. When filled, this tank holds about as much air as fits in a telephone booth. All of that air is compressed down to that small size to fit on your back! Inside a filled scuba tank, there are about 3000 pounds of pressure per square inch, or 3000 psi!

Now, let’s take an empty balloon with us, instead of a glass. We’ll go down to 33 feet and fill our balloon with air. What do you think will happen to this balloon if we take it to the surface? Will it get smaller or larger?

There were 2 atmospheres of pressure at 33 feet, now there is only 1, so the balloon got twice as big. If we filled our empty balloon at 66 feet and brought it to the surface, it would get 3 times as big, or it would pop before we got there!

When you breathe at the surface, you are taking in about a liter of air each time you inhale. If you are underwater with your scuba tank at 33 feet, you are still taking in about a liter of air, but since you are at 30 psi of pressure, or 2 atmospheres, you are taking in twice the amount of air molecules with each breath. The air is twice as dense, therefore, you are pulling air out of your tank twice as fast as you would at the surface. It is now clear that the deeper you go, the less time you have before your air runs out. “

Basic Criteria for Brochure or Presentation: (Formats- Powerpoint/Prezi/webpage/publisher/hand crafted)

Interesting and catchy cover/Title Why Mathematics is essential for Scuba Diving Brief explanation or visual graphic organizer of information above. Include a visual that shows and

explains how many atmospheres of pressure a filled scuba tank has inside. Include an explanation of how deep you would have to go to have the pressure inside a filled scuba tank be equal to the pressure of the surrounding water.

A Chart to show the comparisons of depth, psi, atmospheres and fullness of air in a glass. Read the article, How Deep Can You Scuba Dive? Create your chart to go up to the depth of a recreational diver. You can go deeper if you like. You should include reflections on the patterns that are seen in your chart. With a visual describe what fraction of a glass will be filled with air if you take it upside down to a depth of 49.5 feet.

Balloon Scenarios –5. Describe a balloon with 3 liters of volume at the surface that is taken underwater until its volume

decreased to 1.5 liters and explain how deep it has to be.6. Describe taking an empty balloon (that can only hold 4 liters of air maximum) to a depth of 132 feet

and filling it with one liter of air. Explain whether the balloon will survive the trip back up to the surface or not with visuals.

Big Bob and Little Larry Scenario Cartoon/Comic/Visual3. In this scenario, compare how fast air will be used up when two people are diving. Big Bob, a large

person who breathes at the rate of 1.5 liters of air per breath, is swimming at a depth of 33 feet. He breathes every 5 seconds. Little Larry, a small person, is swimming at a depth of 49.5 feet. He breathes in 0.8 liters per breath and breathes once every 4 seconds.



Radio Sweepstakes

Common Core State Standard(s):6.NS.4


solving them Reason abstractly and quantitatively Construct viable arguments Model with Mathematics Attend to Precision

Materials Needed: Radio Sweepstakes Task Sheet for students. Calculators Technology Access if needed to create presentation and/or visuals Answer Key for Questions included in the Radio Sweepstakes Task

Instructions to Teacher: Students are given a problem-based simulation that involves a radio sweepstakes. Students will be using the concept of finding least common multiples throughout the problem. In this problem the students will compare TWO different ways the sweepstakes can be implemented and must point out the pros and cons of each. They will make a decision as to which way would be the best and must present their findings with full support and detailed explanations of their mathematical calculations. This presentation can be in any style the student wishes, but must be organized and include visuals. A “hint” is given to the students in the task. It is the teacher’s discretion as to whether calculators can be used or not. Students will need a copy of the Student Task Sheet, “Radio Sweepstakes” and access to technology resources if they are available and the student would like to create their presentation with technology. An answer key is provided that answers the essential questions posed in the problem. A rubric for assessment is included. A copy of the student task is below.

Investigation for Students: Radio SweepstakesProblem posed to students:In today’s society, phone-in radio contests are excellent sweepstakes to try to win because the number of entries is much lower than most sweepstakes. Only people in one specific area who are listening at the right time will be able to enter, and a good portion of the ones who are listening won't bother to try to win. This means that the chances of winning are very high.

You work for a new local Radio Station that has $5000.00 to spend in a sweepstakes. The station has decided that the best sweepstakes promotion would be free movie tickets (regularly priced at $7.50) and FREE concert tickets regularly priced at $35.00. The radio station is paying full price for the movie tickets and the concert tickets with its $5000.00.


Your station manager has decided that every 5th and 6th caller will get one free movie ticket and every 20th caller will get a free concert ticket. As soon as a caller gets BOTH a discount movie ticket and a concert ticket, the criteria will change with a consistent pattern. The next callers to get the discount movie tickets will have to be every 7th and 8th caller and the concert ticket winner will have to be every 18th caller. When a caller again gets BOTH a movie ticket and a concert ticket again, the pattern will change to every 9th and 10th caller getting the movie tickets and the concert ticket winners will be every 16th caller.

The pattern continues in this manner until the money is exhausted.

Your station manager feels that this is a brilliant idea. However, the assistant manager doesn’t think the callers will be too interested in just one movie ticket or one concert ticket. The assistant manager’s suggestion is to keep the same patterns of callers but award two movie tickets and two concert tickets.

You have been asked to use your mathematical mind to calculate how many callers will have to be answered and when the sweepstakes money will be exhausted in both scenarios. Your input on how they compare is needed as well as your judgment on which scenario you think is better.

Organize your work as you will have to explain this all to your station manager. You must also argue a case for the scenario you think is better and whether this whole idea is a logical sweepstakes or not. Point out the pros and the cons and offer suggestions that may help. Point out any patterns you see as the sweepstakes continues. Present this in some form to your station manager and assistant manager.

Hints for beginning: (This is the station manager’s idea)Every 5th caller – caller 5, 10, 15, 20, 25, 30, 35, 40…..Every 6th caller – caller 6, 12, 18, 24, 30, 36, …..Every 20th caller – caller 20, 40, 60, ….You see that every 30th caller will include the 5th and 6th callers, so the 60th callers will get movie tickets and concert tickets.Money spent:

for the callers calling in multiples of 5 there will be 12 x $7.50 spent for the callers calling in multiples of 6 there will be 10 x $7.50 spent for the callers calling in multiples of 20, there will be 3 x $35.00 spent (60 is a common

multiple of all three) Total for this pattern = 90 + 75 + 105 = $270

GOOD LUCK!

Sources: About.com – Contests and Sweepstakes Blog.socialmaximizer.com (clip art)


Student Name: ___________________________________________



Insufficient=1

18 points

Progressing=2

21 points

Proficient=3

23 points

Exemplary=4

25 points


Score: _____







Score: _____






Score: ______





Problem Inappropriate Incomplete or Appropriate High level solutions


Solving Approaches and Strategies

Score: _____

with lack of accurate problem solving approaches used.

partially inaccurate use of problem solving strategies.

problem solving strategies used with accuracy and effective analysis shown. Flexibility with strategies is shown.

showing deep understanding of problem solving strategies and flexibility with strategies. High accuracy shown. Synthesizes and evaluates in the process.


Comments:

Investigation: Radio Sweepstakes Answers

This problem involves finding common multiples. The students must present this is some organized way in any form they would like or the instructor assigns them. They must reason as they compare the pros and cons of both scenarios. Their personal judgments of which scenario is better will need sufficient and logical support.

Scenario One (Station Manager): Hints for beginning: (This is the station manager’s idea)

1. Pattern One: Common multiples of 5,6 and 20/ LCM is 60 (How many 5’s, 6’s and 20’s are in 60?) 5’s: 12 x $7.50 =$90 6’s: 10 x $7.50 = $75 20’s: 3 x $35 = $105 Sub- total = $2702. Pattern Two: Common multiples of 7,8 and 18/ LCM is 504 (How many 7’s, 8’s and 18’s are in 504?) 7’s: 72 x $7.50 = $540 8’s: 63 x $7.50 = $472.50 18’s: 28 x $35 = $980 Sub-total = $1992.50 + $270 = $2262.503. Pattern Three: Common multiples of 9,10 and 16/ LCM is 720 (How many 9’s, 10’s and 16’s are in 720?) 9’s: 80 x $7.50 = $600


10’s: 72 x $7.50 = $540 16’s: 45 x $35 = $1575 Sub-total = $2715 + $2262.50 = $4977.50 –this leaves no more money to proceed with another

pattern Total callers IS the LCM for each pattern. Added together, the total callers are 60 + 504 + 720 =

1284 callers

Scenario Two (Assistant Station Manager):

1. Pattern One: Common multiples of 5,6 and 20/ LCM is 60 (How many 5’s, 6’s and 20’s are in 60?) 5’s: 12 x $15=$180 6’s: 10 x $15= $150 20’s: 3 x $70 = $210 Sub- total = $5402. Pattern Two: Common multiples of 7,8 and 18/ LCM is 504 (How many 7’s, 8’s and 18’s are in 504?) 7’s: 72 x $15= $1080 8’s: 63 x $15= $945 18’s: 28 x $70 = $1960 Sub-total = $3985+ $540 = $4525 –this leaves $475 and that will not cover another complete

pattern so students should offer suggestions as to what to do with this leftover money.

Student Investigation: Radio Sweepstakes

Problem:

In today’s society, phone-in radio contests are excellent sweepstakes to try to win because the number of entries is much lower than most sweepstakes. Only people in one specific area who are listening at the right time will be able to enter, and a good portion of the ones who are listening won't bother to try to win. This means that the chances of winning are very high.

You work for a new local Radio Station that has $5000.00 to spend in a sweepstakes. The station has decided that the best sweepstakes promotion would be free movie tickets (regularly priced at $7.50) and FREE concert tickets regularly priced at $35.00. The radio station is paying full price for the movie tickets and the concert tickets with its $5000.00.

Your station manager has decided that every 5th and 6th caller will get one free movie ticket and every 20th caller will get a free concert ticket. As soon as a caller gets BOTH a discount movie ticket and a concert ticket, the criteria will change with a consistent pattern. The next callers to get the discount movie tickets will have to be every 7th and 8th caller and the concert ticket winner will have to be every 18th caller. When a caller again gets BOTH a movie ticket and a concert ticket again, the pattern will change to every 9th and 10th caller getting the movie tickets and the concert ticket winners will be every 16th caller. The pattern continues in this manner until the money is exhausted.

Your station manager feels that this is a brilliant idea. However, the assistant manager doesn’t think the callers will be too interested in just one movie ticket or one concert ticket. The assistant manager’s suggestion is to keep the same patterns


Enter with just a phone call!

Easy sweepstakes! All you need is a

phone!

of callers but award two movie tickets and two concert tickets. You have been asked to use your mathematical mind to calculate how many callers will have to be answered and when the sweepstakes money will be exhausted in both scenarios. Your input on how they compare is needed as well as your judgment on which scenario you think is better.

Organize your work as you will have to explain this all to your station manager. You must also argue a case for the scenario you think is better and whether this whole idea is a logical sweepstakes or not. Point out the pros and the cons and offer suggestions that may help. Point out any patterns you see as the sweepstakes continues. Present this in some form to your station manager and assistant manager.

Hints for beginning: (This is the station manager’s idea)

Every 5th caller – caller 5, 10, 15, 20, 25, 30, 35, 40…..

Every 6th caller – caller 6, 12, 18, 24, 30, 36, …..

Every 20th caller – caller 20, 40, 60, ….

You see that every 30th caller will include the 5th and 6th callers, so the 60th callers will get movie tickets and concert tickets.

Money spent:

for the callers calling in multiples of 5 there will be 12 x $7.50 spent for the callers calling in multiples of 6 there will be 10 x $7.50 spent for the callers calling in multiples of 20, there will be 3 x $35.00 spent (60 is a common multiple of all three) Total for this pattern = 90 + 75 + 105 = $270

GOOD LUCK!


The Great Game Board Company

Common Core State Standard(s):6.NS.4


solving them Model with Mathematics Use appropriate tools strategically Attend to Precision Look for and make use of structure

Materials Needed: Possible graph paper/other paper or poster board Student Task Sheet Calculator Technology Access if used Choice of materials when choosing how to present the boards


Instructions for Teacher: Students are given a problem-based simulation in this investigation where they are being hired to cut and design game boards. In problem solving the different dimensions of the game boards to be cut and the greatest sized squares to draw on the boards, students constantly work with the concept of greatest common factors, GCF. Students must read through the problem and the task and create visuals of the various game boards that they create. Students must include detailed representations of the mathematics used. A sample is given in the student task. Students will need a copy of the student task sheet, “The Great Game Board Company Investigation”. It is the teacher’s discretion as to whether calculators can be used or not. Students will need a copy of the Student Task Sheet, “The Great Game Board Company Investigation” and access to technology resources if they are available and the student would like to create any part of their investigation with technology. There is a final optional task where student pick one of the game board dimensions and use it to create a game for GCF and LCM. This is up to the teacher and how much time there is to spend on the investigation. If students do create a game, their designs for their individual game boards may be hand-crafted using any materials they like or computer generated. An answer key is provided that gives six possible answers for dimensions. Students may come up with more or some that are different. A rubric for assessment is included. A copy of the student task is below.

Student Investigation Problem: The Great Game Board Company has hired you to cut and design their game boards. They are unique in the sense that creating game boards is all that they do. These boards can range in sizes from life size mats to small foldable boards. Different materials are ordered to create the “boards”.

Presently, there is a special buy on synthetic matting that comes in 10ft x 8ft sheets. The Great Game Board Company is purchasing 6 of these sheets.

Task: You must create 6 different game board sizes to be cut from these 10ft x 8ft sheets. For each size that you choose, tell how many game boards can be made from the sheet. Each board must be divided into the largest size squares (measured in inches) that can evenly be

drawn. This will set the board up for its design. For every board you create, you must show the dimensions of the entire board and the

dimensions of the squares. A prototype of the board must be made with all calculations to be submitted for error-checking. You must also show a unique design for each board to be used in a game.

SAMPLE BELOWIn the example below, if one has a board measuring 3ft x 4ft, one can find the dimensions of the largest squares by finding the greatest common factor of the dimensions. 3 feet is converted to 36 inches and 4 feet is converted to 48 inches. The greatest common factor is 12inches so the squares will be 12inx12in.


1 45 65 4 16 232

3

4 feet long – each segment is 1 foot or 12 in 3 feet

Each segment is 1ft or 12 in

Calculations:

36: 2 x 2 x 3 x 3

48: 2 x 2 x 2 x 3

GCF= 2 x 2 x 3

GCF= 12 inches (1foot)

This board was made from a large board measuring 12ft x 9ft. I can cut 9 game boards from this.

A final optional task will be to choose one of your game boards and create a game for GCF and LCM. Your designs for your individual game boards may be hand-crafted using any materials you like or computer generated as the example above. Your final game may be made in any option that you choose.


Student Name: ___________________________________________


Insufficient=1

18 points

Progressing=2

21 points

Proficient=3

23 points

Exemplary=4

25 points


Score: _____







Score: _____







Score: ______





Problem Solving Approaches and Strategies

Score: _____

Inappropriate with lack of accurate problem solving approaches used.

Incomplete or partially inaccurate use of problem solving strategies.

Appropriate problem solving strategies used with accuracy and effective analysis shown. Flexibility with strategies is shown.

High level solutions showing deep understanding of problem solving strategies and flexibility with strategies. High accuracy shown. Synthesizes and evaluates in the process.


Comments:

Investigation: The Great Game Board Company Investigation


POSSIBLE ANSWERS

Students should be showing their work to convert the 10ft by 8ft mat into inches. The entire mat will be 120inches x 96 inches. They should also be showing their work for all of the calculations of the number of boards and the dimensions of the squares on the board. Below are 6 possible answers. The students may come up with a different answer. Make sure it is justified.

1. The largest game board that can be made is 120in x 96in. The GCF of 120 and 96 is 24in or 2 feet. There will be 16 squares on the board and it will be a giant board!

2. The mat can be cut into 5ft x 4ft sizes. This will make 4 game boards. The dimensions will be 60in x 48in. The GCF is 12inches or 1 foot so each square will be 12in x 12in. There will be 20 squares on the board.

3. The mat can also be cut into 5ft x 2ft sizes or 60in x 24in. This will make 8 boards. The GCF is also 12in like the board above. There will be only 10 squares on this board.

4. The mat can be cut into 2 ½ feet by 2feet, or 30in x 24in. This will make 16 boards. The GCF is 6inches. There will be 20 squares.

5. The mat could also be cut into 2feet by 1foot, or 24 in x 12in. This will make 40 boards! The GCF is 12 inches. There will be only be 2 squares on the board however. You would have to be creative to design a game for this one!

6. The mat could be cut into 2 ½ feet by 1 foot, or 30in x 12in. This will make 32 boards. The GCF is 6 inches. There will be 10 squares.

The Great Game Board Company Investigation


Generalization: Concepts of factoring are an integral part of constructing items in the real world.

Problem: The Great Game Board Company has hired you to cut and design their game boards. They are unique in the sense that creating game boards is all that they do. These boards can range in sizes from life size mats to small foldable boards. Different materials are ordered to create the “boards”. Presently, there is a special buy on synthetic matting that comes in 10ft x 8ft sheets. The Great Game Board Company is purchasing 6 of these sheets.

Task: You must create 6 different game board sizes to be cut from these 10ft x 8ft sheets. For each size that you choose, tell how many game boards can be made from the sheet. Each board must be divided into the largest size squares (measured in inches) that can evenly be drawn. This will set the board up for its design. For every board you create, you must show the dimensions of the entire board and the dimensions of the squares. A prototype of the board must be made with all calculations to be submitted for error-checking. You must also show a unique design for each board to be used in a game.

SAMPLE BELOW

For example, if one has a board measuring 3ft x 4ft, one can find the dimensions of the largest squares by finding the greatest common factor of the dimensions. 3 feet is converted to 36 inches and 4 feet is converted to 48 inches. The greatest common factor is 12inches so the squares will be 12inx12in.

A final optional task will be to choose one of your game boards and create a game for GCF and LCM. Your designs for your individual game boards may be hand-crafted using any materials you like or computer generated as the example above. Your final game may be made in any option that you

choose.


1 4

5 6

5

4

16 2

32

3

4 feet long – each segment is 1 foot or 12 in

3 feet

Each segment is 1ft or 12 in

Calculations:

36: 2 x 2 x 3 x 3

48: 2 x 2 x 2 x 3

GCF= 2 x 2 x 3

GCF= 12 inches (1foot)

This board was made from a large board measuring 12ft x 9ft. I can cut 9 game boards from this.

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