Teaching Slope By Building A Linear Ramp and Racing Matchbox Cars

The purpose of this activity is to teach the slope formula by building a ramp for hot wheels/ matchbox cars. Students will estimate the slopes, calculate the slopes as well as test them with their car.

What You Will Need:

* 3-4 Matchbox cars

* 4 Pieces of Styrofoam Board

* 4 Pieces of Construction Paper

* 4 Printable x-y coordinate graphs with a scale of 1-5 for both x and y

* 8 Pencils or Straws

* Material to make the ramp (construction paper, cardboard, etc)

* Glue or Two-sided Tape

Building The Ramps:

It is often times hard to explain how to build things but I decided to use basic materials found in any classroom to make the process easier. If you plan on using this activity every year you may want to look in to building a much more sturdier version of the ramp.

Step 1: Tape the printed x-y coordinate graph to a piece of construction paper (The graph should take up the whole construction paper. You could also draw in the coordinate on the construction paper either vertical or landscape)

Step 2: Tape/Staple/Glue the construction paper from step 1 to the Styrofoam board

Step 3: Make a linear ramp using some types of flat surface durable enough for matchbox cars to slide down. This ramp could be made by cutting in half the cardboard inside of wrapping paper, using folded paper, or wood. It doesn't matter what material you use as long as it is sturdy.

Step 4: Optional: Make siding on your ramp so that the cars slide down without falling off the side

Step 5: Choose where you would like place your coordinate points used to calculate the slope. For example, points (0,5) and points (3,1) for a slope of -4/3

Step 6: Using two sharpened pencils or double sided tape stick each pencil through the construction paper into the Styrofoam (so that both pencils are sturdy) at the coordinate points you desire or use two-sided tape to hold the ramp.

Step 7: glue/tape the ramp onto two pencils

Step 8: Try rolling a matchbox car down the ramp to see if it is sturdy.

Step 9: Make 2-4 different ramps with various inclines and have your students calculate the slope of each

Step 10: Ask each student to guess which slope will yield the fastest speed?

Step 11: Ask each student to guess the various speeds

Step 12: Using a stop watch, have your students calculate the speed in seconds of each ramp.   

Teaching Volume by Packing a Suitcase With Textbooks

Packing a suitcase is an art that I have never mastered. My wife can go on a four day weekend to Miami with a carry-on while I’m stuck checking a large bag in addition to a carry-on. Much of this may stem from my inability to distinguish between what is needed to take on a trip and what isn’t, but it’s also a volume problem. My wife just seems to be able to pack her suitcase better than myself.



The purpose of the activity is to have students calculate the volume of a suitcase, then calculate the sum of volumes they wish to pack inside the suitcase and finally fill it. For this activity I would suggest filling the bag with classroom textbooks (as this tends to be the sum of their worth if you ask me).

You Will Need:

• 2-4 Suitcases

• 2-4 Groups

• Rulers

• Endless Supply of Textbooks

• Worksheet (see types of questions below)

Classroom Activity:

Step 1: Create At Least 2-4 workstations and place different suitcases at each workstation
Step 2: Have students break up into 2-4 groups and give each group one textbook and one ruler

Step 3: Have each group calculate the volume of their textbook and record it

Step 4: Send each group to a workstation

Step 5: Have each group take 5mins calculating the volume of each suitcase at their workstation.

Step 6: After 5 mins, switch stations and record the volume again

Step 7: After the volume of each suitcase has been recorded have each group estimate how many textbooks can fit in each suitcase.

Step 8: Have the groups test their estimation by filling each suitcase with as many textbooks as possible.



Using Student Names To Calculate Areas Perimeter of Irregular Shapes

A fun activity for helping students learn to calculate the area and perimeter of irregular shapes is to have students write their name in a box-type shape and calculate the area or perimeter of their name. The fastest way to expedite the process is to use a special font through a word processor to type each of your students names. You could then make them all the same font size, print them, and cut each name out for each student.

To help you faster determine if each student's work is correct you could write the alphabet in a word processor using whatever font and font size you determine best and then fine the area or perimeter of each letter. This would serve as your key. This way, when a student turns in their work you can compare their score to the score using your key.


Note: This activity will take a lot longer if students have to measure each side of each letter in their name. Thus, this would work well for a homework assignment. However, if you want to use this lesson in class. You might want to have the measurements provided ahead of time, or, have students round to the nearest whole number.

Running Shoes Vs Basketball Shoe

The purpose of this post is to test the performance of a running shoe versus a basketball shoe. We will document the difference in weight and test their performance by running a quarter of a mile Sprint. Do the running shoes, in fact, make you faster. If so, how much faster? Faster by what percentage?

You Will Need

• A stopwatch

• One pair of designated basketball shoes

• One pair of designated running shoes Note: Both shoes much be the same size.

This activity will be much easier if you, the instructor, have a pair of basketball shoes and running shoes. However, most of us might have one of the two but not both. And some of us may not have either. No worries, some of your students will have both (those that run track/cross country as well as play basketball) or you can always ask your local shoe store to donate/discount them for your classroom. Needless to say, you want to start preparing for the project weeks in advance.

Preparing For The Project:

1) I would start asking your students who has a pair of both running and basketball shoes at least two weeks in advance to see what types of resources you have at your disposal. The project works best when you have 4-5 teams, each team with 1 runner.

2) After you have solidified your 4-5 runners, split up the rest of the class into teams. Each team will have 1 runner.

3) Designate a track or field to run.

4) Create a Project Paper For Each Team (See below)

Project Worksheet For Each Team:

The types of questions you ask will be dependent on the skill-set/grade-level of your students. I will give you a rough idea of the type of worksheet I would create but you may need to critique it for your students.


Before Activity:

Team Name_______________________

Names Of Students In Team______________________________________________

Weight of Basketball Shoe______________ (can be found online or by using a scale)

Weight of Running Shoe ________________ (can be found online or by using a scale)

What is The Difference Between The Weights of Both Shoes_____________________

What is The % Difference Between The Weights of Both Shoes___________________


Activity:

Time For Lap One Basketball Shoes___________________________________

Time For Lap Two Running Shoes_____________________________________

Difference Between Two Running Times_______________________________

Percentage Difference Between Two Running Times______________________


Additional Questions:


Would you say the running shoes have a significant impact on a runners speed?


If not, why might runners still decide to purchase a running shoe over a basketball shoe?

Back In The Classroom:

Now that your students are tired of running and screaming it is a good time to go over the results from each team. Compare the percentage differences of each team; both of the running time as well as the weights of the two shoes. If you would like, write the statistics on the board and have your students calculate the averages of all the teams.

Discuss the last two questions that were posed to the students and ask how marketing plays a role in selling shoes. You may also want to ask the runners which shoe felt more comfortable while running or they thought the times would be different if they were running a longer race?

Homework Questions:

1) Given the percentage difference in the two running times, how much time would a runner save on their 8:13 mile if they were to use these running shoes as opposed to the basketball shoes they used?

2) Would you pay $113 for these shoes? Yes No

Tree Cutting Activity Using Mathematics

The purpose of this activity is not actually to cut down any trees, this is a job for a skilled professional, but simply to mimic the types of cuts many professional tree cutters use in order to open up a dialog with students about how math plays a purpose in tree cutting.

This exercise will involve students going outside to measure various sizes of trees using rulers, tape measures, proportions etc and then discussing how said trees should be fell. For example,what direction should such a tree be fell? What angles should be used in our cuts to ensure the tree falls in that direction? What might happen the if the angles of the cuts were to be altered.

You Will Need:

Groups of 3-4

Tape Measure (preferably the type used to measure waste lines so that it can easily wrap around the diameter of a tree)

Pen & Paper

Worksheet With Questions (Provided Below)

Types of Cuts For Falling a Tree

I own a landscaping company and I've cut down quite a few trees in my lifetime but falling trees is not my area of expertise. Instead we are going to focus our concentration on two different types of cuts and leave the details to the experts. Remember the purpose of this exercise is to get your students thinking about how to follow mathematical directions and what role mathematics plays in various professions.

What Your Students Should Be Able To Do

1) Student will take measurements of the trees diameter and label the tree small or large

2) Students will determine which cut is best to use based on the diameter size

3) Student will draw a picture of the tree on a piece of paper with a diagram of obstacles that surround the tree.

4) Student should estimate the height of the tree

5) Student should choose the direction the tree should fall and mark it on their diagram

6) Students will label where the cuts will be made on the tree to ensure the correct fall

7) Students should know the depth and angle of each cut and be able to label them

Definitions:

Big Tree (Diameter larger than 9 inches)

Small Tree (Diameter less than 9 inches)

Cuts Involved in Falling a Small Tree

• Determine the direction of fall

• Cut 1, on the side where you intend for the tree to fall. The cut should be parallel to the ground about ¼ of the way through the tree.

• Cut 2, on the opposite side of the tree where the first cut was made make your second cut (labeled a back-cut). Make this cut downward at an angle of 30 degrees.

Cuts Involved in Falling a Large Tree

• Determine the direction of the fall

• Cut 1: Labeled the Top-Cut, should be made on the side of the tree that you wish it to fall. The top cut should (under normal circumstances) be parallel to the ground, about waste high, The cut should be about 1/3 of the trees diameter.

• Cut 2: Labeled the Bottom Cut, should be made 4-8 inches below the 'Top Cut', at an upward angle approximately 20-30 degrees. Continue this cut until the 'Bottom Cut' reaches the end of the 'Top Cut' creating a pie shape piece labeled the The Notch

• Cut 3: The Back Cut, should be made on the opposite side of 'The Notch', at the same height of the 'Top Cut'. Cut at slope horizontal to the ground.

**Note, this post is not in any way meant to prepare anyone to cut a tree. It is meant to help kids see what math loggers use everyday within their cuts and how slight error rates can cause major accidents. This article does not provide near enough information and what information that is provided is representative of only certain scenarios. The information is taken from the following websites. Here, Here and Here.

Teaching Proportions Using Ant Trivia

Proportions are as fun a subject to teach as they are important to learn. Armed with the knowledge of how to setup and solve proportions, students can tackle all varieties of real life phenomena. The purpose of this post is to add some variety to your proportions by introducing trivia about ants.

The majority of the trivia about ants is taken from the following website 

 Trivia Fact 1:

It is estimated that at any time there are more than one quadrillion (1,000,000,000,000,000) ants on the planet.

Possible Proportion Question:

If there are one quadrillion ants for every 7 billion humans, how many ants per human?

Trivia Fact 2:

The average life span of an ant is 45 to 65 days.

Possible Proportion Question:

How many total ant lifespans are equal to one human lifespan of 80 years?

Trivia Fact 3:

Ants are able to lift things 20 times their own weight.

Possible Proportion Question:

If an ant can lift 20 times their own weight how much could a human, with the strength of an ant, lift if the human weighted 150 pounds?

Trivia Fact 4:

An ant brain has about 250,000 brain cells. A human brain has 10,000 million

Possible Proportion Question:

How many ants would it take to equal the number brain cells in one human brain?

I will be creating a worksheet based on this Ant exercise sometime in the future.  

Helping Students Understand A Zero Numerator vs. A Zero Denominator

Students normally have trouble remembering solution to problems involving a zero numerator or a zero denominator. Most tend to remember that the solution is either zero or undefined but they tend to mix them up. The reason for this is students are relying on memory vs comprehension. Students deserve an explanation and sometimes a simple true or false question can help.

Let us take a look at the following two fractions

0/8

and

8/0

As educators, we know that zero divided by eight is equal to zero (0/8 = 0). Likewise, we know that eight divided by zero is undefined (8/0 = undefined), but why? Let me offer an informal explanation using Pizza and then I will comment on some other proposed explanations.

Take a look at these eight slices of pizza and ask yourself the following two questions.

Is it possible to eat zero of the eight slices of pizza?



Is it possible to eat eight of the zero slices of pizza?


These two questions, although not formal, should offer a strategy to students trying to understand why they cannot divide by zero but may divide into zero. Create a worksheet with different pictures of food (pizza, pie, etc) and ask these two questions for each problem. Soon your students will approach each problem dealing with a zero numerator or a zero denominator with this mindset.

Additional Comments:

 

Others on the internet have offered similar informal reasoning to help students distinguish solutions to fractions with a zero numerator or a zero denominator. For example, one website offers the following reasoning in relation to separating objects into piles. The argument follows something like


8/0 is undefined because you can't separate 8 items into 0 piles.

0/8 = zero because separating zero items into 8 piles will still equal zero piles.

Although I think this is a clever way of thinking about fractions with a zero numerator or a zero denominator, I'm not sure it would stick well with students. Students may begin thinking that since it's not true that you can separate zero items into eight piles and that it's also not true that you can separate eight items into zero piles thus since both are untrue, both are undefined.


Others have sought to show the illogical nature of dividing by zero by setting up an equation and showing the properties of equalities

8/0 = x



Multiplying both sides by zero yields

8 = x (0)

or

8=0

Although I do think this is the best method for demonstrating how the properties of equality do not allow division by zero I'm not sure your students will remember to set up such an equation if testing. Thus, I think coupling this method with an explanation like that above would be the best fit.

Congruent vs. Similar Shapes While Appealing To Fashion Senses

The goal of this activity is to help make the distinction between the two concepts of similar shapes and congruent shapes. We will do this by appealing to our students fashion senses.

 

The Hook:

Do you guys think other people in this school copy your style? Give me an example without using any names?

Transition:

What about shoes? Has anyone ever went out and purchased the same pair of shoes you bought or wanted? Now lets be honest, did they purchase a similar pair or an exact pair? Were they the same style, color, etc? What about the same size? So would you say this person purchased the exact pair of shoes you wanted or a similar pair?


In math we use terms like this when dealing with stuff like shapes. Except we don't say similar or exact, we say similar or congruent.
For example,
These are similar triangles

These are congruent triangels

Whats the difference?
So let's return to our example of shoes. If we (informally) accept that similar shoes are the same shoe but in different sizes than what would congruent shoes look like? Same shoe, same size!

Classroom Exercise:

Everyone take out a piece of paper and draw a line down the middle. Label one column Similar and the other column Congruent. I want each of you to go around the room and see who has similar or congruent shoes, jackets, shirts, pants, jewelry, etc. Anything but socks or underwear! We'll keep it PG for this exercise!

Math Field Trips Part 1: Mini Golf

As I've posted before, why should the arts and humanities get all the fun for field trips? There are some great field trips within reach of any school. Although I have mentioned mini golf before in previous posts, I want to post it again with some additional ideas of how to make it work.

Mini Golf!

Putt Putt offers a multitude of math lessons you need only draw the students attention to the game play.

Tally The Degrees & Angles:

Have students keep a tally sheet of the various types of angles or degrees they use to hit each shot. This could be done in numerous ways depending on the skill set of your students.

• For younger students, you could have them tally the number of Straight, Right, Obtuse & Acute angles they use in each shot (for example, putt one was a straight line, putt two was an acute angle, putt three was a straight line in the hole)

• For older students you could have them sum the angles used at each hole (for example, putt one was a straight line 180degrees, putt two was a 20 degree angle that went into the hole. The sum of degrees was 200.

• For even older students you could have them sum the degrees of not only the initial hit but also the degrees of angles after contact (for example, putt one was a straight line 180degrees, that took a 45 degree bounce off the wall and stopped).

Use Negative & Positive Score Counts:

Each hole will have a designated par. For example, hole one might be a par three, meaning the golfer should be able to make it in the hole in three shots. To introduce negatives, have each student start at the par and count backwards for each stroke. Thus, starting at a par 3, a student who took 5 shots would have a score of -2. Therefore, in contrast to traditional golf scoring, the highest score wins.

Additionally, in teams of three, each student could tally the scores at the end of the hole and rank them in order from least to greatest.

No Putt Putt Course Within Driving Distance? Make One In Your Classroom!

It doesn't take a contractor to make something for your students, they will have a blast no matter how cheesy it is. Simply bring in a couple boxes, some putters, golf balls. Take one of your wide boxes that is not very tall and cut a hole in it. Use the other boxes as obstacles. If you need a boarder bring in some 2x4's. Take some pics and send them to us and we will post it on our page!



Teaching Proofs & Formulas Part 2: Middle School

(Click For Cool Pi Pictures and Shirts)

 

The Circle & Pi

What is a circle? What is Pi? What is the relationship between a circle and Pi? Students use Pi everyday with little-to-no understanding of it. To a student, Pi is simply a little symbol that pops up every now and again. Sometime they get to shorten it and write it as 3.14, sometimes they are asked to use it as a fraction 22/7. They apply it when asked, but it has little to know value to them. My philosophy is to generate value by taking it's value away from them. Let them operate in a world without the availability of Pi and see if they begin to value it.

Lets start with the formula for the circumference of a circle

C = (D)(pi)

or

C=(2)(Pi)(r)

An educator can state the relationship of Pi as the quotient of a circles circumference and its diameter. over and over again but the student will tend to neglect the relationship. To remedy this we are going to have students calculate, by hand, the quotient of 10 circles' circumferences to their diameter. In other words, a student is going recalculate an approximation of Pi over and over and over again ten-fold until they start to pick up on a trend.

All Aboard The Pi Train!

What You Will Need:

 

1) A Printout of 8-10 Circles of various sizes and label them by number

2) A ruler

3) A tailor style tape measure (or a bendable ruler to measure circumference)

4) A recorder sheet for answers

For Example: Circle 1 Circumference_______in. Diameter________in. C/D=______

Circle 2 Circumference_______in. Diameter________in. C/D=______

Etc.

Steps:

Step 1: Separate students into pairs (individually works as well)

Step 2: Hand out materials

Step 3: Have Students begin to measure each circles circumference and diameter and record them on the answer sheet

Step 4: Have students calculate, by hand, the ratio of each circles circumference to its diameter.

Step 5: Ask the students to write what trends they have noticed.

Classroom Discussion:

After everyone is done with the exercise, discus what each student just did.

Examples of Questions

“What did you get for the C/D of circle 1? circle 2?”

“Did anyone get anything less than 3 or bigger than 4?”

“Which circles C/D was the closest to Pi?”

“Why do you think this is”

“Does anyone believe they measured their circle perfectly?”

“What would you need to measure it perfectly?”

“If it was measured perfectly and calculated perfectly what would you get?”

Recap Of Lesson:

The biggest idea that you want your students to walk away with is that Pi is something special. It is defined as a circles circumference divided by its diameter (See note below regarding this). Just knowing this bit of information would have allowed your students to have written “Pi” as their answer in the above 10 problems and they would have been right each time (without all the extra work). Now we need to think of a thought experiment to solidify the value to your students.

Thought Experiment:

Try this,

“You are a captain within a 9th century medieval military. Your primary role as head of the archery units is to protect your kings small castle from outside invasion. Your castle has become surrounded by an army of barbarians. You know that your archers can accurately shoot clusters of enemy targets from 100 yards, but you do not know the distance from your men to the targets standing along the entrenched moat. However, you do know the distance all the way around the outside of the circular moat is 600 yards. Can your archers hit their target?

C = (2)(Pi)(r)

We don't know (r) but we know C

600 = (2)(Pi)(r)
dividing 600 by 2 yields

300 = (Pi)(r)

dividing 300 by Pi roughly yields

95.5 yards = radius

Thus, if your archers are standing at the center of the circle, can they hit their target at a radius of 95.5 yards away?

Based on the assessment made earlier, yes!

Homework Question:

Do you think the majority of your archers are located at the center of the circle, closer to the enemy or further away? Why?

User Note To Appease The Math Gods: Although obvious, given the title of this post, to not offend anyone, let me point out that our definition of Pi is limited to Euclidean Space. More formal definitions of Pi are not applicable.

Prime Number Factorization Hangman (Updated)

This Worksheet Is Free To Download. Look On The Right-hand Side of The Webpage Under "Math Downloads". It will be labeled "Prime Factorization". Best of luck! 

Teaching Place Value While Standing In Line

Everyday you must line your students up before going outside, going to lunch, etc. Why not use this time to teach place value? All you need is a good system. Take a look at this place value system aside of this line of students and you should get a pretty good idea of what I have in mind.

Now, the trick to this exercise is how to 'place value' a classroom of 25-30 students. Unless one is inclined to venture out to the trillions or trillionths, it would probably be best if the instructor used multiple lines or one large single-file line with multiple systems of place values.

 For example,

let us assume you have five rows of five student desks in your classroom (of which I will show two methods) depending on how you line up your students.

1st: How do you line up your students? Do you call them by row? By name? All at once?

If by row: I would suggest labeling each desk tens, tenths, etc and calling them up in one or two rows at a time. Leave it up to the students figure out which one of them should be first, second, third, etc. You could also hand out index cards with names such as “tens place” and a pic of where the “tens place” is located.

If by name or all at once: I would either hand out index cards with names such as “tens place” and a pic of where the “tens place” is located or I would label the floor next to the door (or wall) with place value names and tell them their place value as you call them up.

For example,

“Lets get ready to line up to got to lunch”

“We will line up based on name and place value, the biggest place value is lined up first”

“Billy and Stacy you line up first, Billy, you are the “tens place”, Stacy you are the “ones place”.

--This forces Billy and Stacy to discuss who is first in line--

“Joey and Amy you are next line. Joey, you are in 'tenths place', Amy you are in the 'hundreds place”

Note: If your students have never been introduced to the place value system, you cannot expect them to know how to arrange themselves. Thus, a poster on the wall may serve well