## Friday, November 30, 2012

### Teaching Integer Addition & Multiplication Tables Using Corn-hole

Corn-hole has become all the craze for parties. I can think of a dozen different ways they could be used to teach math lessons. I will share two of these ideas below.

Using Corn-hole to Teach Multiplication Tables

This activity would be great for the whole class but I truly think it would be a wonderful tool for helping those students who are behind to catch-up. If I was still in the public schools, I would single out the students who still struggle with their multiplication tables and find time during activity period, study hall, etc for them to play this game.

Essentially, you will have two different colored bean-bags. Both colored bean-bags will be labeled 0 through 12 (this can be done with a black marker). The student will be given both sets of bean bags. They will choose two bags from both colors and toss them.

Rules of Play

1) If they make both colors in they will multiply them and add this number to their score

2) If they make one color but not the other, they multiply them but do not add them to their score

3) If they miss both of them, they multiply them and subtract the number from their score.

Examples Of Three Tosses:

Toss 1: Blue Bag 2, Red Bag 5

-I make both: I record my answer and my score is 10

Toss 2:Blue Bag 6, Red Bag 1

-I make one, but not the other: I record my answer but my score is still 10

Toss 3:Blue Bag 4, Red Bag 2

-I miss both: I record my answer and subtract 8 from my score (new score is 2)

Score Sheet:

A score sheet could be something simple like

Red Bag #_____ Blue Bag #_____

_____x _____ = _______

Circle One: Made Both Missed One Missed Both

Score______

Using Corn-hole to Teach Integer Addition & Subtraction

The setup is the same as above, two different colored bean-bags. Both colored bean-bags will be labeled 0 through 12 (this can be done with a black marker). The student will be given both sets of bean bags. They will choose two bags from both colors and toss them.

Only now the blue color represents positive integers and the red bags represent negative integers.

After the toss, students will record the sum of the positive and negatives.

Examples Of Three Tosses:

Toss 1: Blue Bag 2, Red Bag 5

-I make both: I record my answer as positive 2 combined with negative 5 (2-5 = -3) and record my answer as a positive score. My score is 3

Toss 2:Blue Bag 6, Red Bag 1

-I make one, but not the other: I record my answer as positive 6 combined with negative 1 (6-1 = 5) but nothing is added to my score. My score is still 3

Toss 3:Blue Bag 4, Red Bag 2

-I miss both: I record my answer as positive 4 combined with negative 2 (4-2 = 2) and subtract this from my total score. My score is now 1.

## Wednesday, November 28, 2012

### Tips For Teaching Horizontal & Vertical Shifts

To help your students understand vertical and horizontal shifts in graphs they need to start thinking in terms of x/y-intercepts, not x/y values. We will label this the method the horizontal slide vs the vertical slide.

Horizontal vs Vertical Slides of Function Graphs

Here is the graph of f(x) = x^2

Here is the graph of x^2 with a Vertical shift of 2 units (f(x) = x^2 + 2)

Here is a graph of x^2 with a horizontal shift of 2 units f(x) = (x-2)^2

Students' Trouble In Understanding

Most students tend to understand vertical shifts. It seems intuitive to them that adding 2 to x^2 will shift the graph 2 units in the positive direction. However, students tend not to understand the horizontal shifts. It seems backwards to them. The reason for this is that students are concentrating on what is being done to the variables as opposed to the x/y-intercepts. The task of this activity is not mastery but to shift the students' focus to what's happening with the intercepts instead of the what is being done to the variable.

What You Will Need

• graph paper
• multiple color markers
• Activity Sheet

Steps:

Step 1: On blank (x,y) coordinate graphing paper have students plot the following graph by generating random points.

f(x) = x^2

Step 2: Have students analyze the graph and determine the x-intercept and the y-intercept.

Step 3: Ask them what would you need to do to the graph of x^2 to change the y-intercept.

Step 4: Have them redraw the graph of x^2 anywhere else they want on the y-axis as long at it doesn't shift to the left or right.

Step 5: Ask the question how many units did your graph shift upward or downward?

Step 6: Have them contemplate what their new function will look like. Will it be x^2 plus 2, minus 2, multiplied by 2, divided by 2, etc.

Step 7: Show them what the new function will look like

f(x) = x^2 + 2

Step 8: Have them determine what the function would like if their graph what shifted up 2 more units. What would it look like if it was shifted down 5 units?

Step 9: Have them draw the two new graphs and write the new functions beside them.

Step 10: Ask them what changed in the graph, what remained the same.

Repeat the process with Horizontal shifts, having them concentrate on the x-intercept as opposed to what is being done to the variable x

## Tuesday, November 27, 2012

### How Much Soda Do You Loose From Fizz When Pouring?

Intro:

You know the routine, you open of a 16 ounce bottle of Pepsi and start filling your glass. As you pour the soda you notice a fizz is starting to form causing you to stop and wait so the carbonated bubbles don't spill over the rim of your cup. While waiting you begin to wonder, how much soda am I loosing by pouring it? Maybe I should just drink it from the bottle, at least I wouldn't need to wait for the fizz. The purpose of this activity is to figure out how much, if any, soda you loose when pouring it.

What You Will Need:

• Multiple 16 ounce bottles of soda (one for each group)

• A measuring cup that measures 16 ounces (16 ounces = 2 cups)

• Activity Sheet (below)

• Paper Towels

• An intro speech about being careful to avoid spills
Outline of Activity

Students will split-up in groups. Each group will pour 16 ounces of soda into a measuring cup (WITHOUT SPILLS!). They will wait for the fizz to halt and continue pouring until all soda has been poured. Students will then estimate how much, if any, soda was lost during the pouring stage and then express this number a percentage loss. Note, there will certainly be a margin of error in this activity, as most measuring cups are not accurate and there will be some amount of soda left in the bottle. However, these will serve as great after-activity discussion questions.

After Activity Discussion:

How much soda did each team loose during the process?

Why do you think soda was lost when pouring?

Is this enough loss to avoid pouring?

Does anyone know what creates fizz—be sure to ask your science teacher?

How accurate do you think your estimates are?

What could be throwing off your results?

How much soda do you think is left in the bottle?

Homework:

Estimate how much soda your family looses per month from pouring soda?

Activity Questions:

1) How much soda﻿ is in the bottle?__________

2) How much soda, if any, was lost during the pouring?_____________

3) Express this loss as a percentage_____________

## Monday, November 26, 2012

### Math O’clock

Whether you’re teaching addition, subtraction, multiplication, division or algebra, nothing is better on a Monday morning than getting your kids out of their seat to do math problems. I call this game Math O’clock.

Steps:

Step 1: Have 12 students form a circle like a clock with each student representing an hour, like so.

Step 2: Have the rest of the class form a line

Step 3: Using index cards with pre-chosen problems that yield answers between (0,12], hand each student in line a problem in the index card for them to solve.

Step 4: Students must demonstrate their solution by lying on the ground inside the circle-clock using their feet as the hour hand and their arms as the minute hand to demonstrate the solution to their index card, like so.

Step 5: Make sure you have a rotation schedule so that all your students get to play.

## Wednesday, November 21, 2012

How long would it take a ball to drop to the ground if you reached as far up as possible? Let's find out with some help from quadratics and the square root property.

What You Will Need

• Golf balls, tennis balls, etc
• Stopwatch
• Measuring tape
• Formula for acceleration due to gravity neglecting air resistance.
• Activity sheet

Overview and Goal of Activity

The goal of this exercise is for students to use algebra to solve a real-life application. Students will work in pairs (or groups depending on the amount of stop watches you have at your disposal) to calculate how long a small ball will take to fall from the tips of their toes, reaching as high as possible, to the ground. Students will first use algebra and then will test their answer by actually timing the fall

Steps:

Step 1: Using adhesive or tape, tape at least one tape-measure to the wall from the floor to the ceiling.
Step 2: Pair up students and give each group a stop watch and an activity sheet (below)
Step 3: Have students measure themselves in inches or centimeters and record their height—Make sure they are measuring from the tips of their toes to their finger tips while reaching to the ceiling.
Step 4: Next, using the formula s(t) = 16t^2, where t is their height in seconds have them plug in their height for s(t).
Step 5: Have students do their best to solve for t.
Step 6: Have the students record their answer
Step 7: Have students go back to their seats
Step 8: Explain the process of how to solve for t by using the square root property.
Step 9: Have them partner back up and solve for t and record their answer
Step 10: Have students test their answer by dropping a ball and timing it. Have them record their answer
Step 11: Back to their seats for after activity discussion

Example of Activity Sheet

1) How tall are you from your toes to your finger tips __________cm/inches

Formula for acceleration due to gravity is
s(t) = 16t^2

2) Plug in your height for s(t) and do you best to solve for t

t = ___________

4) Using your notes from above, plug in your height for s(t) and solve for t

s(t) = 16t^2

t = __________

5): Using your stop watch take turns dropping the ball from the tips of your toes and record you answer (do this at least twice).

Time for ball to hit the ground ___________seconds

After Activity Discussion:

Why do you think there might be a difference in the times?
What does the s(t) stand for in the equation?
What does the 16^2 stand for?
Why do you think the 16 is squared?
Do you think your time would be different if you dropped a heavy rock instead of the ball?
Do you think your time would be different if you dropped a leaf?
Why do we ignore the minus in our two answers?

Homework:

Actual Problem Worked Out

Height = 84inches

84 = 16t^2
84/16 = t^2
21/4 = t^2
+/-squar root (21/4) = t
t= +/-2.29 seconds
t = 2.29seconds

## Monday, November 19, 2012

### Student Lockers & Math

The purpose of this activity is to provide insight into the math behind a student’s locker. The main topic covered is that of permutations, however, we will refer to it as combination since this is what students refer to it as). Even if your students are not ready for permutations they will find the activity interesting and it may serve as a great introduction to other math topics.

Imagine that you wanted to steal a diary out of your best friend’s locker to see what she was writing about you. You don’t know her locker combination but you plan on guessing three times a day until you get in. Will you eventually get into it by the end of this school year?

Let’s assume her lock has 40 digits and her actual combination (PERMUTATION!) is 3 digits. Take a guess how many different possible combinations there are? Next, calculate how many guesses you will be able to make at three times per day within a school year (180 days x 3 = 540 guesses).

The actual number of possible combinations (PERMUTATION!) calculated is 64,000.

With this number in mind, calculate how many days you would need to ensure you guessed the right combination (64,000/3) =21,334 days)?

What percentage chance pre year is this (540/64,000= 0.8%)?

The math behind the number?

The following is know as a Permutation with repetitive numbers allowed

Each digit has a 1/40 chance of being correct. Thus, with a 3 digit combination possibilities are calculated as

40 x 40 x 40 = 64,000

Activity Sheet:

1) Will you eventually get in to her locker by guessing three times a day? _____

2) How many possible combinations there are?

3) How many guesses will you make at three per day for a school year?

4) Based on this, with 540 guesses what percentage of chance do you have at guessing the right combination? ______%

5) Given the actual number of combinations, how many days would you need to ensure you guessed the right combination?________How many school years is this_______?

6) What is your actual percentage chance of guessing the right combination in a school year? ___________

## Friday, November 16, 2012

### 10 Ways To Make Your Math Class More Fun

I often get emails from educators asking how they can make their math class more fun. My normal suggestion is relocate your class to Toys-R-Us, but, alas, if this option is not available, here are at least some suggestions.

1) Mini Lessons:

Divide the total number of minutes you are in class by 5; this is how many lessons you should aim for. Thus, a 60 minute class should have approximately 12 lessons. For example, a 12-part mini-lesson class about fractions may look something like this.

• Mini-Lesson 1: Warm-up by asking what the class would rather have 2/3 of a pizza or 6/7, why?

• Mini-Lesson 2: Draw two circles, one with three parts, one with seven parts, and shade them. Then ask the same question as before and see if their answers differ

• Mini-Lesson 3: Have students label each part of their circle 2/3 or 2/7 and cut out each piece

• Mini-Lesson 4: Pair-up and have one student give their partner 1/3 of one pizza and 4/7 of the other. Have the other student give their partner 2/3 of one pizza and 2/7 of the other and then have them compare who has more pizza

• Mini-Lesson 5: Have all students return to their seat and close their eyes. Read two fractions aloud and ask them to raise their hand for the one they think is larger. Tally up the results and place them on the board. Do it again for another two fractions.

• Mini-lesson 6: Using the previous examples, show why one is larger than the other

• Mini-lesson 7: Discuss how knowing this might help them in real life, ask for their input

• Mini-lesson 8: Have your class form a circle while holding hands with you in the middle. Show them how ½ a circle is different from 1/3, ¼, or 1/5 by standing in the middle and extending two tape measures

• Mini-lesson 9: While still holding hands, take turns asking the students to guess different fraction sizes by reading out each others names “1/3 would be from Julie to Charlie, ¼ From Julie to Jamie, etc”

• ---I could go on forever but you get the point
Keeping it short will hold your students attention.

2) Get Them Out Of Their Seat:

Sitting for an hour straight is hard for me and I'm a 32 year-old man. In kid-time that's even longer. Switch it up! Form a circle like you did for story time in elementary school. Let them spread out on the floor. Turn out the lights and give them flashlights to do their worksheets. Sound dumb? Not as dumb as the expectation of having kids be attentive while seated for an hour.

3) Art:

Using art is a great way to illustrate word-problems. In fact, have your students draw pictures before answering any problem and see how much more attentive they are to the solution. Example, asking 5 x 3? Ask them to draw 5 apples, then another 5 apples, then another 5 apples.

4) Legos:

Using just a few Lego pieces will allow you to teach almost any concept. You could calculate volume, area, build polygons, build irregular shapes, calculate ratios, probabilities of one color, etc, etc.

5) Posters:

Got some boring definitions you need your students to learn? Have them create a poster. Better yet, tell them they are opening up a store and their only item for sale is this definition and therefore they need a sales-poster!

6) Group Work:

Group work can be a great learning tool. However, it's often barely used or underused. You need to be specific about what you want each member of the group to contribute. Solving an equation? Assign each member a task. Person A can only combine like terms, person B can only add or subtract, etc.

7) Less Problems & More Mastery:

Two awesomely thought-out problems are worth more than an entire worksheet of drill. There may be a time for lots of problems, for example the multiplication tables, but these are few and far between. Using just two problems will allow you to create content. Make your students value to the solution by actually creating a problem they would want solved. For example, why would the death/kill ratio be a better indicator a players skill in Call of Duty than simply number of kills? Or, the most text messages ever sent in a day is ____ what would their cell phone bill be if they had to pay 0.03 per text? Lets create an equation and solve it.

8) Homework Should Be Short and Sweet:

I hate the idea of homework. I hate the idea of taking time away from family and friends to do busy work. If you must use homework make it something really fun. Have them explain to their parents over dinner how a mathematician won \$100,000 for finding the next largest prime number and if their parents can't remember a prime number have them explain it to them. Have them research the largest number they can find and bring it back with them the next day for you to write in scientific notation. Give them an equation to solve and ask them to draw a picture of a guy drawing a picture of a guy solving it...I don't know! Just give them something fun

I know our establishment is anti-calculator but I assure you this, my two home-schooled boys will not be doing silly math problems by hand, I don't even care if they know their multiplication tables. I will put a calculator in their pocket and we will go outside and shoot bottle rockets to discuss math. My suggestion is thus, let the calculator do more of the tedious work and use your skill as an educator to concentrate on the why questions.

10) Use Math Stories, Math Trivia or Riddles

We all love a good story or a good riddle. Take some time to discuss how math is used today or how battles were won due to a generals mastery of mathematics, etc. Post weekly riddles or obscure equations on your walls. Tell them stories about quirky mathematical geniuses of the past. Find ways to peak their interest in the subject.

## Wednesday, November 14, 2012

### Using A Two-Column Proof To Argue With Parents

I've always enjoyed books on logic. As opposed to pulling my hair out while studying it in college, I once attended an online seminar with David Gordon of the Mises Institute titled “How To Think: An Introduction To Logic”. I can't recall the text we used for the course but it was a fun class. Anyone suffering through an elementary geometry class has, at some point, witnessed a two column proof in which the left hand side is used to denote a proposition while the right hand is either an axiom or something that can be derived from a former proposition (sometimes a hypothesis as well). Like so,

The purpose of this exercise is nothing more than introducing the two-column proof by allowing students to create a method for arguing with their parents. My advice would be to hook the class by asking, "what is something your parents won't let you do? Has anyone tried arguing with their parents? I mean really arguing with them and showing the flaw in their statement? Let me teach you"

NOTE: The outline below is nothing more than a fun, informal introduction. It's merely an attempt of playing with the process of creating a proof, the arguments presented below will be like that of a young child.

The argument presented is as follows. You want a cell phone but your parents claim you can't have one because you don't have enough money to pay the bill and you might run-up their bill (you could also use you are not old enough). How are you going to prove to them that this statement is not true. Or in essence, how can you prove that you do, in fact, have enough money for a cell phone.

## Monday, November 12, 2012

### Mathematics & Eyesight: Part 1 Field of Vision

I've always loved learning about social-biology and the effects evolution plays on development. One area of particular interest is the eye. More specifically, why do some animals have eyes sight that allots them a larger field-of-view than others? Why can some animals see farther or more acutely but have less peripheral vision? This activity incorporates degrees of a circle for better understanding. It would be great to couple this lesson with a biology teacher on your team. Note: The degrees used here are rough estimates and may be subject to “Google Error”. Feel free to double check the suggestions and send me corrections. The activity will still work the same.

What You Will Need

• Paper & pencil

• A 360 degree protractor

• Activity sheet (see below)

Overview and Goal of Activity

The activity will involve students looking at pictures of different animals and guessing their field-of-view. Students will then be told their field-of-view and trace the degrees of vision using a 360-degree protractor. After sketching the degrees of vision students will take guesses as to why this would help the animal survive.

Steps:

Step 1: Hand each student a 360-degree protractor and activity sheet (see below).

Step 2: Our first animal is the horse

Step 3: Have students guess at the degrees of vision of a horse

Step 4: Provide them the actual degrees of vision of a horse and have them sketch where they think the blind spot is (see below).

Step 5: Provide them with the actual information of a horses blind spot (see below).
Step 6: Have students guess to why such vision would be beneficial to a horse.

Example of Activity Sheet

1) Take a guess at the degrees of vision of a horse ____________degrees

2) Why do you think this? ______________________________________________________________________________________________________________________________________________________________________

3) The actual degrees of vision of a horse is near _350___degrees

4) Using your protractor, trace 350 degrees and where you think the blind spot is.

 (Notice the blind spot in the front)

5) Why do you think horses developed this form of eyesight?

Continued Activity

Step 7: Repeat the same steps for each of other animals you wish to study

Examples
Human Field-of-view = nearly 180 degrees

Owls Field-of-view = 110 degrees

Dogs view =250 degrees

After Activity Discussion:

Why do you think animal have different degrees of vision? Why do most hunter have a more narrow, forward degree of vision whereas most non-hunters have more peripheral view?

Homework Question:

What would life be like if your degree of vision was like that of a horse?

## Friday, November 9, 2012

### Brand Name Paper Towels...Are They Worth It?

“Bounty, The Quicker Picker-upper....Bounty!”

How good are the brand name paper towels really? Do they really hold more liquid than the off-brand paper towels? What is the price difference? What is the unit price difference? Are they worth the investment? Let's examine them and find out.

What You Will Need

• 1 roll of a name brand paper towel

• 1 roll of an off-brand paper towel

• The price paid for both

• Measuring Cup

• Kitchen weigh scale

Steps:

Step 1: Demonstrate both types of paper towels (to the class) and have them estimate their cost

Step 2: Provide the actual prices of both types and have your students calculate the difference in price (problem 1 on activity sheet)

Step 3: Have students calculate the percentage difference in price (problem 2 on activity sheet)

Step 4: Pass around both rolls of paper towels around the classroom and have students indicate how many actual number of towel sheets are within the roll. If they are 2-ply have students double the number listed (problem 3 on activity sheet).

Step 5: Have students calculate the unit cost of each brand of paper towel. (problem 4 on activity sheet)

Step 6: Ask which brand of paper towel is a better deal in terms of unit price. (problem 5 on activity sheet)

Activty

Step 7: Break the class up into groups. Provide each group a piece of paper towel from both brands.

Step 8: Explain that we are going to see how much water each paper towel can absorb by weighing it.

Step 9: Ask the groups to spread out on the floor (hopefully the floor is not carpet)

Step 10: With your measuring cup, spill the same amount of liquid in two spaces on the floor beside each group (thus, each group will have two spills, each for both paper towels)

Step 11: Have students absorb as much liquid without it dripping from both spills (the towels cannot be dripping as they are walking).

Step 12: When ready have the students bring the (non dripping) paper towel up and weigh it (or bring the scale to them). I would suggest one paper towel at a time.

Step 13: Have the students record the weight in ounces. (problem 6 on activity sheet)

Optional Step 14: After complete, provide the weights of each type of paper towel with no absorption (meaning, using new pieces from the roll, weigh both types of paper towels). Feel free to skip this step if the weight of the paper towel is negligible.

Step 15: What is the difference in weight between both paper towels? What is the percentage difference? (problem 7 on activity sheet)

Step 16: Given the amount difference in price and the difference in absorption which do you think is a better deal? (problem 8 on activity sheet)

After Activity Discussion:

What made you decide on the better deal, the price or the absorption rate or both? Do you think you would change your answer if you owned a day care with lots of spills per day?

Homework Question:

Estimate how many spills you have per month in your household ____. How many rolls of paper towels would it take you to wipe up that many if you used one roll as opposed to the other.

## Wednesday, November 7, 2012

### Deconstructing 3D shapes to 2D Shapes. Making The Connection

It is a common theme that at some point in time students in a math class will create 3D shapes from 2D pieces of paper. This is a great exercise for students and I would encourage anyone to do it before they begin a lesson on volume or surface area (it also makes for a light homework assignment). However, the purpose of this exercise is to deconstruct 3D shapes to properties of 2D (see note below about 2D/3D) to show the connection between both dimensions.

What You Will Need:

• Pre-assembled 3D shapes (cereal boxes, cubes, or square based pyramid will work just fine)

• tape

• Something to cut with (either scissors, or a childproof utility knife)

• Groups work best if you don't have a lot of 3D shapes

• You could also ask the students to bring in a shape from home made out of cardboard

Goal:

The goal of this exercise can be two-fold, depending on the skill level of your students. At the minimum we want students to recognize that their 3D shapes are composed of 2D shapes. However, if you wanted to couple this exercise with surface area you could have them calculate the surface area of the 3D cube or square-based pyramid prior to deconstructing it and then calculate the sum of areas of the 2D shapes after it is deconstructed.

After Activity Q & A

Why is it helpful to think of 3D shapes as 2D shapes

What is the difference between the 2D shape and the 3D shape

What is the difference between the formulas for each

Is the 2D shape really 2D or does it have a small height?

Are there any 2D shapes?

What would a 4D shape look like

Note Regarding 3D & 2D

It may or may not be true that an actual two dimensional object can exist, after all, even atoms are composed of three dimensions. However, this is not the point of the exercise. For example, are shadows, electrons, TV pictures, etc, 2D? These are great questions for a physics forum or post-project discussion but beyond the scope and purpose of this lesson. The purpose is to connect the concepts of 2D and 3D in a manner that grade school students can understand

## Monday, November 5, 2012

### Having Students Blueprint Their Own Parking Lot

We park in them daily but do we ever stop to ask what really goes into the process of designing parking lots? Clearly some white chalk and measuring but anything else? Yes, quite a lot actually. Most parking lots are a product of careful planning by land surveyors and business owners attempting to maximize space while abiding to local zoning laws. Do we use perpendicular or angled parking? What happens if the lot is oddly shaped? How do we insure everyone can safely enter, park and exit the lot with as little backup and congestion as possible? I can think of no better way to find out then to have your students design a parking lot.

You Will Need:

-Groups of four

-Four types of pre-drawn parking lots (you could make them square, rectangle, Pentagonal or even oval). I would use either heavy construction paper or poster board. See pic
﻿
 Example
﻿

-Rulers

-Pencils

-Markers

-Four match box cars to serve as a model

The Goal:

The goal is for students to maximize the amount of parking spaces within their parking lot while also providing lanes for the cars to enter and exit the parking spaces and the lot and then to discuss what math they used, could have used, or used without them noticing.

Steps:

Step 1: Split the class up into 4 groups and give each group the needed supplies listed above

Step 2: Assign each group a pre-drawn parking lot.

Step 3: Have each group come up with the type of business they are opening and create a business name as well.

Step 4: Explain to the groups that the goal is to maximize the amount of parking spaces within their parking lot while also providing lanes for the cars to enter and exit the parking spaces and the lot.

Step 5: Have each student measure the length and width of their matchbox car and explain that each parking space must account for the width of opened doors.

Step 6: Students should use a pencil to begin with before using a marker

Step 7: The goal at this point is to provide as little structure as possible and let students discuss strategies among themselves.

After Project Discussion:

After the work is completed the instructor should ask why each group decided on their design and if they think they maximized their parking spaces for their business. After group discussions a classroom discussion could be held on how each group could have bettered their design using different angled parking spaces. See below curtsey of RWCG

﻿

## Friday, November 2, 2012

### Making a PB&J Using Rational Equations

Rational Expressions and Rational Equations can be tough for students of any age to grasp, let alone if they are nestled within a word problem. The goal of this exercise is to connect the students with the material by engaging in a real-life problem...making peanut butter and jelly sandwiches (denoted PB&J henceforth).

What You Will Need:

--Peanut Butter

--Jelly

--Two plastic knives

--Two volunteers

How It will Work:

Two volunteers will be timed while each making a peanut butter and jelly sandwich (see below for what constitutes completion). Two make the problem less tedious, have your students record times in seconds as opposed to minutes. Thus, a student who completes a PB&J in 1min and 15 seconds should have a recorded time of 75seconds. After each students has their time recorded, students will calculate the amount of time it should take both volunteers to make one PB&J. After calculated, a test will be performed with both volunteers working together to make one PB&J.

What Constitutes A PB&J Sandwich:

What seems obvious is not. We must have a benchmark for what constitutes a completed PB&J in order to compare the two times. We will say that a PB&J is complete after

3) Both pieces of bread are stacked on top of one another

4) The lid for both the jelly and the peanut butter have been sealed back on

5) The knife has been washed/wiped off.

Steps:

Step 1: Choose Two Volunteers

Step 2: Explain what each volunteer will do and what constitutes a finished PB&J sandwich.

Step 3: Have first volunteer make a PB&J and record their time.

Volunteer One Can Make a PB&J in ______seconds

Step 4: Have second volunteer make a PB&J and record their time.

Volunteer One Can Make a PB&J in ______seconds

Step 5: Have the class calculate how long it should take both volunteers to make one PB&J if they work together.

Step 6: Work out the proposed solution for your students on an overhead (see example below)

Step 7: Test the proposed time by timing both volunteers making one PB&J

Worked Out Example

Recorded Time Of Volunteer 1: ____75______seconds

Recorded Time Of Volunteer 2: ____80______seconds

Combined Time Should Be

1/75 + 1/80 = 1/x
LCD = 1200x
1200x(1/75 + 1/80) = 1200x(1/x)
16x + 15x = 1200
31x = 1200
x = 39 seconds (rounded up)