A Fancy Way of Adding Zero & The Additive Identity

I have a love/hate relationship with riddles. I love them when I can solve them quickly, hate them when they keep me up at night for something that is normally too obvious for me to notice.

Here is an older riddle, if I can call it that, which your students may love.

“Take any number. Now add 5 to it. Multiply that by 2. Subtract 6. Divide by 2 and finally subtract 2 and you back to your original number. Why does this work?”

1) Let your students try it first on a couple of different numbers

2) Now let them try to figure out why it works on their own.

3) Have them write down the whole operation, paying special attention to order of operations. (((((x + 5) * 2) – 6) / 2) – 2)

4) Again, let them try to figure out why it works on their own.

5) Ask them to go through the following operation ((((5 * 2) – 6) / 2) – 2) and see what number they get (zero is the answer)

6) Ask them to restate the Additive Identity

7) Thus, this whole problem is simply a fancy way of adding zero to your initial number.

In Pursuit of Perfection of Form

Our students often never understand how they misplaced a negative symbol or added instead of subtracting. They often think of this is a part of the math process, and it is, to an extent. Many times, to my own embarrassment, I've stood in front of a classroom and butchered a problem on the white board. My only self-medication for these "duh" moments was knowing that we deal in an unforgiving science.

But still, I often wonder why our students are so sloppy in their work. I once pulled a young man, who was an incredible basketball player, off to the side to ask how he perfected is form so well in the sport. His response was of course practice. If I would have thought about it at that time I would have paralleled his form in basketball with his form in my lecture hall, but I didn't.  I'm writing this now so that you may draw this parallel with your students.


Analyzing this

May be the key to helping your students perform this 

Coffee Taste Test Using Same Filter (Blind Sampling & Data Collection)

A question occurred to me today. As normal, I need my afternoon coffee. However, feeling excessively lazy, instead of making a new pot of coffee, I decided to simply fill the coffee pot back up and use the same filter from earlier this morning. As you can imagine, it wasn't exactly indulgent. Yet, I began to wonder if the coffee I was drinking was actually weaker, or was it my anticipation of it being weaker that made it so. Project!

Have your students conduct a blind taste test with other faculty members being their dummies. I would even let your students call them dummies...at least for the day. The taste test will involve each faculty to sample 3 small cups of coffee and record their data without knowing which cup of coffee is which.

Cup 1: First pot of coffee.

Cup 2: Same filter, same coffee, 2nd pot of water

Cup 3: Same filter, same coffee, 3^rd pot of water

The survey should be something simple, but measurable. For example, have them rank each cup of coffee on a scale of 1-10 with 10 being the highest.

Things to account for

1. When ranking, will the dummy be comparing the tastes of one cup to the other cups only, or will they be comparing them to previous cups of coffee. For example, if the dummy loves a White Chocolate Mocha from Starbucks as their number 10, they may rank all three of your cups a 1's or 2's. Thus, to account for this, you may want to raise the ranking system from 1-10 to something like 1-100.
2. Will the cups be served black or with sweater and cream? If the latter, you shuld make sure that each cup receives nearly the same amount.

After Results:

Have each student calculate some descriptive statistics (mean, median, mode, standard deviation, etc) and draw some conclusions. Does using the same filter and same coffee really make the coffee noticeably weaker? Or is it in our imagination?

Finally, the most important part of the activity is the after-activity discussion. Why might the tests be inaccurate? Was are sample size appropriate? What could we have done better? How might a double-blind test affect the results? Why might companies use blind taste tests when comparing their product to their competitors

Have fun...send me some pics if you decide to do the project!   

Representation & Modeling: Getting Your Students To Draw A Math Picture

A simple picture can go along way in helping your students solve an application problem. This is as true in pre-algebra as it is in calculus, however, representing the problem with a picture or diagram is often an overlooked step by students. The goal of the blog post is to help students see the value.

Drawing a diagram or other type of visual representation is often a good starting point for solving all kinds of word problems. It is an intermediate step between language-as-text and the symbolic language of mathematics. By representing units of measurement and other objects visually, students can begin to think about the problem mathematically. Pictures and diagrams are also good ways of describing solutions to problems; therefore they are an important part of mathematical communication. From Teacher Vision

Activity:

Take one full class and introduce nothing but word problem with the emphasis on using pictures to capture the problem. Next, quiz students on this vary concept. Read aloud to them a word problem and have them represent it in a picture while at the same time labeling everything they know. Use some variation the following rubric when grading their work.


Quality of Representation 1 2 3 4 5

Proper Labeling of Parts 1 2 3 4 5

Unneeded Detail             1 2 3 4 5


Lets Call Upon Another Example From The Teacher Vision Article

Question: A frog is at the bottom of a 10-meter well. Each day he climbs up 3 meters. Each night he slides down 1 meter. On what day will he reach the top of the well and escape--From Teacher Vision

Here is a possible representation of this problem.

Using Geometric Formulas To Simplify Life.

Bring in a nicely wound cord or garden hose (see below)

Explain to your students that you want them to determine the length of the cord or garden hose with the only condition being that they cannot stretch it out; they must find another way to measure the length.



Notice that the wound cord or garden hose is simply a bunch of circles stacked on top of one another. What if we calculated the circumference of the top circle and multiplied it by how many circles are formed by the winding process? Certainly you will be dealing with a margin of error here but it will at least get you in the ball park.

Note: be careful not to count a circle more than one time.  

Area of Irregular Shapes Using Roman Tortoise Formation

Out of curiosity, I decided to estimate the how effective the Roman Tortoise Formation was against opposing archery fire. Having only a few minutes during my morning coffee, I decided to limit my estimation to simply the front of the Tortoise Formation. Such an exercise would be fun for a classroom as well. Here was my thought ‘quick-an-easy’ process, feel free to point-out something more scientific or any potential mistakes.



The Goal: To estimate the area prone to archery fire with the Tortoise Formation.

Given: Google tells me that the Roman shield Scutum had the following dimensions


Height of Shield: 42 inches

Width of Shield: 26 inches.

Average Height of Roman man was 5 feet, 6 inches

Converting to feet, here is a general idea of the shape from the reference point of an archer standing directly in front of the formation.

Noting the rectangular shape and multiplying the average height by the width of six Scutum shields gives us an approximate area of

5.5ft x 13ft = 71.5ft^2

A = 71.5ft^2

The obvious areas prone to archery attack are the legs below the shield and within the semi-ellipsis surrounding their heads. To estimate these areas we could subtract the height of the shield from the average height of the man (5.5ft – 3.5ft = 2ft) and multiply this by the width of the six Scutum shields.

2ft x 13ft = 26ft^2

L = 26ft^2

As for the area’s within the six semi-ellipsis surrounding their heads, we could use the width of the concave shields (2.17ft) as half the circumference of the ellipsis and approximate the height of the radius as .5 feet. This would give us an approximate area of 1.7 ft^2 per ellipse (multiplied by all six would yield 10.2ft^2).

E = 10.2ft^2

Thus, for an opposing archer standing in front of the Roman Tortoise Formation would have approximately

• Area Prone to Attack = Area below the shield (L) plus semicircle areas above their shield (E).
• Area Prone To Attack = L + E
• Area Prone To Attack = 26ft^2 + 10.2ft^2
• Area Prone To Attack = 36.2ft^2

Or, approximately 51% of their body is exposed.

**Note: This seems a little high so I might have made a wrong approximation in length somewhere. Also, the roman soldiers body does not encompass the entire exposed area, so one would need to account for that as well. Nevertheless, this was just for fun.

Incorporating Puzzles (Homeschooling Our Two Boys)

Puzzles are excellent brain training and co-ordination improvement tools and are quite fun! In particular, they develop your abilities to reason, analyze, sequence, deduce, logical thought processes and problem solving skills. These types of puzzles also improve hand-eye co-ordination and develop a good working sense of spatial arrangements. See Full Article

Exponential Pushups: How Many Can Your Students Do

The purpose of this exercise is twofold; first, to show exponential growth, second to serve as a fun way of testing your student’s knowledge of exponents.

• Ask you classroom who can do 2 push-ups? Have someone show you.

• Explain that this is the same as doing 2^1 pushups.

• Ask you classroom who can do 4 push-ups? Have someone show you.

• Explain that this is the same as doing 2^2 pushups

Continue this exercise until you get to 2^7, from here have them calculate and image doing that many pushups. Where you stop from here is up to you.

Next, pair up your students and give them a worksheet with exponent problems. The only catch is, they must show you what the answer is by doing pushups.



Example,

• 3^2 a student from the group must show you the answer is 9 by performing 9 pushups.

Which Letters Of The Alphabet Are Graphs of Polynomials?

This light exercise is to get your students thinking about polynomial graph behavior. You could extend this exercise to guess they degrees and signs of the leading coefficients if desired. Simply print out a worksheet of all the letters of the alphabet and ask your students which letters are graphs of polynomials based on the following distinctions

1. Polynomial graphs are continuous. You can draw them without lifting your pen
2. Polynomial graphs have no sharp corners or cusps, they are smooth (see pic below).
   
   
   
   
   
   

(From www.mathisfun.com)

See mathisfun.com for a great tutorial and pics.

Using Word Search For Learning Key Words

Let's assume you want your students to learn all the key words for addition, subtraction, division or multiplication. Try using a word search without telling them the words to search for.

The Power of Actually Counting Large Numbers

One Thousand Pennies

 

Most four-year olds are very inquisitive. Mine loves developing his number-sense. He loves to ask me “Daddy, which is bigger ___ or ___ “as if the two numbers were about to bout, with the larger being the victor. Without ever practicing counting, he has learned to count to roughly 100 with only a few mistakes. More interesting than this is his development of number placement. Whereas many younger children can count, say to 10, ask them to count backwards and they begin to struggle. Although not fast, my son tends to count backwards just as well as forwards and I think the reason for this is that he developed his sense of numbers slowly without being asked to chant “one, two, three, four…” out loud until he remembered it. With this in mind I want to show you where counting has much more power, with large numbers.

One day my son asked me why 1000 was larger than 100. As most would, I started by telling that it takes ten 100’s to make 1,000, etc, etc. But noticing the blank look I asked for his patience as I decided to count aloud to 100 and then continued onward to 1000. His bright eyes sold me! There was no doubt after about 15 minutes later that 1000 was bigger than 100. To this day he still remembers how big it was.

 

How To Use This In A Classroom

For a better appreciation of larger numbers from your students have them count them.

Take 15 mins of down time and have your students listen as you count aloud to 10, than 100 than 1000.  Have them time you (you will most likely average one number per second). Next, have them calculate how long it would take to count to 10,000 or 100,000 or 1 million.  Have them express this in terms of days or hours.  

One Million Pennies

One Billion Pennies

 

 

Purchasing Fictional Stocks For Math Class

The goal is for students to calculate values & percentage change of fictional stocks. One could easily have them graph results as well.

Getting students interested in investing and budgeting can start in your math class. Give each student $100 of play money and a copy of stock/commodity prices (easily found in Wall Street Journal or online). Have students scour the handout of stock prices and pick which stocks/commodities they would like to invest their $100 in. After doing so, have them calculate how many shares they purchased of each (round to the nearest tenth for ease).

Next, over the course of the next month, have them track their investment (either by you reproducing the pricing sheet or by them checking online). Each week have them recalculate their balance based on the change in stock price and the percentage change.

Example:

Change in Value:

$100 @ $25 per share = 4 shares.

If after 1 week the new price is $24 per share your investment is = $96 (4 x 24).

Percentage Change:

96 – 100 = - 4

- 4/100 = -.04 or – 4%

At the end of a designated time period, have students sell their shares and pay them back in pretend money.

**Differentiated Instruction: If you had some advanced students who really liked this exercise but wanted more, you could discuss short-term capital gains taxes and have students calculate their actual profit.  

Measuring Repulsion & Attraction With A Ruler

I love magnets! Even as a kid I enjoyed carrying magnets with me everywhere I went. I still remember the first time I stuck a magnet up to my grandmothers TV (Opps!). Although the study of magnetic properties can be pretty involving, magnets themselves offer great ways to bring simple lessons to life. In this lesson we will measure the effects of repulsion & attraction when two magnets interact. We will measure their interaction with a ruler.

What You Will Need

• Rulers

• Various Size/shape magnets (preferably labeled A, B, C...)

• Activity Sheet

Overview

Students will use various magnets to measure repulsion and attraction. Using a ruler, students can use the same polarity to repel the other magnet and record the distance of that repulsion. Next, using the same two magnets, students can use opposite sides of polarity to measure the distance along the ruler from which two magnets attract one another (be careful with the fingers!)

Steps

Step 1: Create groups of partners and give each group a ruler, bag of magnets and activity sheet.

Step 2: Have groups spread out along the floor

Step 3: Have student pick two magnets and use a ruler to measure the distance upon which they attract, and repel.

Step 4: Filling in activity sheet

Activity Sheet

1) Magnet______ and Magnet_____

a) Attraction begins slightly at _______cm apart.

b) Repulsion begins slightly at _______cm apart

c) When set beside each other they repel ____cm distance

d) At ______cm the two magnet fully attract (snap together)

The Ultimate B-Day Expression

The purpose of this activity is to have some fun doing expression work. Students will substitute information into an expression with the end result being their birthday. Feel free to use calculators to make this a quick exercise. Here is the expression.

X = Month Of Your Birthday


Y = Day Of The Month Of Your Birthday

Note the vast amount of grouping symbols the kids will need to keep track of.

Example: My birthday is November 21st

x = 11
y = 21

My Math is thus

7(11) = 77
77 - 1 = 76
76(13) = 988
988+ 21 + 3 = 1012
1012(11) = 11132
11132 -11 -21 = 11100
11100/10 = 1110
1110 + 11 = 1121
1121/100 = 11.21 (11.21 is my b-day)

Firewood By The Truck-load: Volume & Per Unit Pricing

Let's assume you got behind on your firewood splitting and needed to purchase some firewood for a week till you get caught up. You check you local paper and see they have prices of $80 per truck-load. What can one expect from a truck-load of wood? How many units? How much is this per unit? We will be using some basic volume to get the number of units and use this number to get a per unit price. We could then decide how long it will last based on types of wood, etc (Note: The purpose of this activity is for the application more so than for the usefulness. As anyone whose ever split wood knows, quality has more to do with density and how well the wood is seasoned..see below)

Measurements Used

Full Size F-150 Long-bed Truck

Length = 8ft

Width = 6.5ft

Height = 1.6ft

Piece of Firewood (We will assume the average log length is 18inches and has a radius of 8 inches which will be cut into 8 pieces for firewood)

Units of Firewood by Volume

For the sake of ease, let us assume that a truckload of firewood will be neatly stacked but not higher than the sides of the truck.

Volume of truck bed

8x6.5x1.6 = 83.2ft^3

Volume of Firewood Log

Changing Our Units to feet

length = 18 inches = 1.5ft

radius = 8 inches = 0.66ft

Volume of a cylinder equals

V = (pi)(r^2)(h)

V = (3.14)(0.44)(1.5) = 2.05ft^3

Volume of Firewood Piece (once split into 8 pieces)

2.05ft^3 / 8 = 0.26ft^3

Units of Firewood Per Truckload

Now that we know the volume of both the truck-bed and for a piece of fire wood we can simply divide them to see how much firewood should stack neatly into the truck-bed

83.2 / 0.26 = 320 pieces of firewood

Adjusting For The Odd Shapes

Since the pieces of firewood are not cubic, they are circle-sections. They will not stack neatly together. There will be added spacing between them. The seller could compensate for this by stacking the firewood above the walls of their truck, but we will assume they didn't.

This spacing could be compensated with a little optimization. But, from stacking wood I think it would be safe to say that four every 10 pieces you will loose one to spacing. Thus we will loose 1/10th of our total to spacing. 1/10 of 320 = 32

320 – 32 = 288pieces of firewood.

Prices Per Unit

At the rate of $80 per truck-load. We have $80 per 288 pieces of wood

80/288 = $0.28 per unit.

The Calculations Are Quite Arbitrary

These calculations are quite arbitrary in their usefulness. Most firewood comes in different sizes. A true calculation of how good the deal on firewood would to compare volume to density as well as how seasoned. The purpose of this posting was simply to serve as an application.

From Here We Could

From here we could ask questions like.

1) If we use 40 pieces of firewood per night, how long will this last us?

2) If you wanted to, you could make a step-function based on how cold it is outside.

At 40 degrees we use 10 pieces of firewood per night

At 30 degrees we use 30 pieces of firewood per night

etc.

3) Could recalculate based on the volume questions for different size truck-beds or different materials

Addressing Our Probability Inefficiencies

The purpose of this activity is short but important. That is to help students NOT associate number of outcomes with probability of outcomes. One could probably teach this to their students faster than they can read this post.

The Problem:

Khan Academy (the wonderful, wonderful people they are) was asked by Lebron James what the chances of making 10 free throws in a row? I won't attempt to outdo Sal on this one

LeBron Asks: What are the chances of making 10 free throws in a row?: LeBron James asks Sal how to determine the probability of making 10 free throws in a row

But I want to draw attention to what I find to be a more interesting problem. That is, why do students tend to associate the number of outcomes with the probability of outcomes? In other words, students tend to sometimes think of outcomes such as hit/miss, win/loose, yes/no, etc in terms of each result having a 50/50 chance. Or again, we tend to think that probabilities of outcomes are always distributed equally. This is a dangerous error to make in life, but the good news is that this is often more an academic mistake than a real-life mistake

For example, ask the following two questions to the same person and see what answers you get.

1) If you shoot a basketball, whats the chances of it going in?

2) If you shoot a full court shot, are you more likely to make it or miss it?

The Solution:

The good news is that you can quickly teach kids to be skeptical of this by taking them to a basketball hoop during activity time or gym class and asking the two questions above, then testing them.  

Variation Of "Bowling For Facts" Integers Addition

I love the creativity of elementary teachers! One notable project I found was First Grader At Last 

I will reproduce their idea below and then show you a variation of how it could be used for teaching higher level math.

Here is a recap of their project

From First Grader...At Last

“Today during math stations, the kids went bowling for addition facts...without the real pins and bowling balls, but still pretty fun! This is a partner game, and each player gets a bowling mat and a handful of colorful chips (I call them mini bowling balls). The players take turns rolling 2 dice, then adding the sums of the numbers rolled. When one of the players rolls a number found on one of their pins, they "knock over" that pin with a chip. The first player to have all pins knocked over wins the game! “

Here is a variation of the game that would allow you to teach adding and subtracting integers.

Using red and white dice (red =negative numbers, white = positive numbers)

The new bowling mat would look something like

New Bowling Answer Sheet For Integers

Example

Roll Both Dice

Red = 5

White = 2

-5 + 2 = -3

Polygonal Glasses & Superhero Masks

Need a fun way for students to learn different polygonal shapes? How about making some special polygonal glasses or masks? Notice the octogon shape lenses?

(Hey, no laughing at my horrible artwork!)

Today were going to expand our attire…polygonal style! The purpose of this activity is for groups of students to make their own masks or glasses while using different shaped lenses.

Here is a link to some printouts.
http://www.firstpalette.com/tool_box/printables/superhero.html

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.

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