Warm-up

If a wood duck can fly at an average speed of 50mph, how far would it take the duck to fly 2043 miles if they only flew 8 hours per day?

Warm-up

If ducklings swim at a rate of 0.9 feet per second, how long will it take them to go a mile?

Making Your Thinking Visible

It is important to help students learn to make their thinking visible in math because it helps students become aware of their own understanding and thought processes and those of others.

When students share their thinking, it means ideas are formulated, refined, and adjusted through sharing.  We see visible thinking all the time when people brainstorm, collaboration, and discussions.  It is an important skill to have in today's world.

In the mathematical classroom, visible thinking is the key for students to learn math and be successful.  We can observe visible thinking in the math classroom during discussions, drawing, writing, and any way that conveys ideas.  We see it when teachers share their thinking with students, when students explain their thinking, when they listen to others explain their thinking, when they discuss their ideas, when they read for understanding, when they write down their thoughts, and when they are able to demonstrate their thinking through the use of technology, manipulatives, or drawings.

There are some things we can do as teachers to help students learn to make their thinking visible. Unfortunately, by the time I get them in high school, most of my students prefer just answering questions without making their thinking visible so I have to work on this.

At the beginning of a unit, have students do the "See, Think, and Wonder" exercise where they preview the unit and answer the questions - "What do you see?", "What do you think you'll be learning?", and "What do you wonder about this Unit?"  This allows students to begin thinking about the unit.  This can be done as a warm-up either verbally or via technology.  If you use technology, you can get answers from others.

When introducing a problem, ask students questions like "How do you think you might solve this?", or "What are you confused about?", or "How can you confirm or counter what you are thinking?", questions to help them think about the problems, rather than just learn by rote.  Further more add a vocabulary activity such as Chalk talk where the teacher writes vocabulary words on paper and hangs them around the room.  Give each student a marker where they can write down a definition, or comment about the term without talking to each other.  If you prefer, set up a google form that allows them to do this.  A second use for the Chalk talk activity is to use it for student reflection.  Write questions on individual sheets of paper so students can go around and write down their understandings, examples, or clarifying questions.

As you work through the unit, ask students "What makes you say that?" during discussions to help them clarify their own thinking. It can be used after the student makes a prediction, or a claim, or even a statement.  Students have to explain their reasoning to show how they arrived at the prediction, claim, or statement.  Unfortunately, too many books write problems in such a way as to follow the particular process being taught in the section, rather than providing more open ended problems.

For instance, if the section is looking at proportions, a student might see a problem that tells them that  Jimmy walks 1mile in 10 minutes while Jane walks 1 mile  in 12 minutes.  If Jimmy lives 1.5 miles from school and Jane lives one mile from school, who arrives home first?  What if you wrote the information that Jimmy walks 1 mile in 10 minutes while Jane walks 1 mile in 12 minutes down and the two children live between one and five miles away from school, then asked questions such as if Jane arrives home first, how far might the houses be apart?  If Jimmy arrives how first, what might be the distance between the two houses? If Jimmy and Jane arrive home at the same time, what is the shortest distance the houses might be apart?  What is the farthest distance apart?  In the second problem, the questions require more thought to answer.

One final activity to wrap up the section is the "I used to think........ but now I think......." because it offers students the opportunity to acknowledge how their thinking has changed over time about a topic or concept.  Often students are not given time to see how their thinking on a particular topic or concept has changed from the beginning to the end of the unit.

It is important to slip activities in to help students make their thinking visible. These are a few ways you can do it. Let me know what you think, I'd love to hear.  Have a great day.

The Conway Knot - Solved!

It was recently announced that Conway's knot problem has been solved.  This particular problem has been unsolved for about 50 years but a young graduate student successfully determined it was not a slice.

Conway's knot problem involves a knot with 11 crossings.  It was named after John Horton Conway and is similar to the Kinoshita - Terasaka knot.  The question about this knot involved  being able to slice the knot.

This problem of knot theory is actually a branch of topology.  It looks at the nature of the spaces in something resembling tangled loops.  This particular area of study has contributed to the understanding of DNA, the behavior of economic markets, to even the universe.  They've been looking at both one dimensional knots and the behavior of two dimensional knots in four dimensional space.  In mathematics a knot does not have two distinct ends, its ends are connected and cannot unravel.

The question involved with the Conway knot boiled down to "Is the Conway knot what is left after a knotted spare is sliced?"  Is the Conway knot a slice?  In other words, can it be untangled but some knots are so crumpled, they cannot.What made the Conway knot stand out is that Mathematicians were able to figure out the answer for all sorts of knots with 12 or fewer crossings except for this particular 11 crossing knot.

The grad student, Lisa Piccirillo heard about the problem back at a conference on low- dimensional  topology and geometry back in 2018.  It piqued her curiosity so she decided to work on it outside of school as if it were a homework problem.  She had a new technique in mind she wanted to try and it worked.  In about a week she had the answer and when she shared it with one of her professors who said it had to be published.

The proof is based on creating a complicated knot that is trace sibling for the Conway knot. She did this because it is known that trace siblings have the same slice status.   Then she applied the Rasmussen's s-invariate to the trace sibling and showed it was not a slice therefore the Conway knot is not smoothly slice.  This means that it is a slice of a crumpled knot but not a smooth one.  Her proof was nice and short and elegant.  It lead to her receiving and accepting an  offer from MIT for a tenure track position.

Unfortunately, John Horton Conway just passed away on April 11, 2020 from COVID-19 at the age of 82.  He contributed so much to the field of mathematics during his lifetime.  Did he learn about this solution to the Conway knot?  I don't know but I am sure if he did, he'd find it quite intriguing.  Let me know what you think, I'd love to hear.

The Futures Market

 The futures market is rather interesting because a buyer and seller enter into a contract to purchase/sell a commodity such as corn at a specific price on a set date.  The seller is hoping he is able to get more than the going price while the buy is hoping to end up paying less than everyone else.

We see references made in television shows when someone comments about pork bellies futures or corn but the truth is that many business use this technique. For instance, airlines might sign a contract with a fuel company to provide so much fuel at a certain price on a set date.  This way the airlines knows exactly what they will pay while the fuel company has a guaranteed sale.  On the other hand, there are people who participate in futures contracts to make money rather than actually take delivery of the product.

Futures contracts are usually carried out at a exchange-traded market which is the intermediary between the two parties.  Basically the contract has the two people agreeing to a price K that is agreed to today but to be delivered at a later time T.  K refers to the futures or strike price while T is the delivery or maturity time.  The buyer or the one who agrees to purchase the item is said to be long while the seller who sells the item is said to be short.

The way it works is that the buyer contacts a a specialized broker asking for a contract for a certain amount at a certain price such as 6000 bushels of corn to be delivered in September.  The broker passes the order to a trader associated with the Chicago Board of Trade asking for one order for 6000 bushels of corn.  The Chicago Board of Trade is an exchange specializing in the buying and selling of commodities and it was established back in 1848.  Originally, it only traded agricultural items such as corn or soybeans but over time has grown to include gold, silver, Treasury bonds, and energy.

At the same time, someone who has 6000 bushels of corn available, contacts a broker to offer it up for sale.  This broker contacts a trader on the Chicago Board of Trade to sell this and the two traders talk, enter into a contract for price and delivery.  Often times, the person buying the commodity will sell it off later, before they take delivery on it.  Many people use this as a way to increase their value without relying on the stock market.

There are some activities you can do in the classroom to help students understand more about futures markets.  The first one, talks about a commodity challenge before the actual activity.  Ignore the contest because it is from 2014 but the activity is worth doing.  It has students take an unopened bag of M & M's.  The students predict how many of each color are in the bag before they open it.  Based on this prediction, the students then decide how many they want to presell at a certain price before calculating how much they make.

Then they open the bag and sort through, determining how many of each color they actually have and they only write down the number they have for each color they didn't presell.  If they oversold a color, they can't count it so have to use a zero and at the end of this step, they calculate how much they make on the ones they didn't presell.  At the end, they determine how they fared overall.  It has questions for students to answer after they have worked through the activity.    It includes all the instructions and worksheets you need to do it in the class.

The second one has a lovely presentation explaining the history of futures, defines everything about futures, the types of people who are involved in futures, information on going long or short, and just about everything you need to know about the topic.  In addition, there is a note sheet to accompany this presentation for students to fill out as they watch it.  At the end, is a quiz students may take to show what they've learned.

Finally is this manual put out by Chicago Mercantile Exchange for high school teachers of agriculture.  The eight chapter book explains everything step by step and includes mathematical examples of everything so a student can actually understand there is math involved every step of the way. In fact the eighth chapter specifically focuses on the math involved with cash marketing, futures, and options math.  The examples are quite clear and the practice problems are well done.  At the very end, one can find all the answers.

Although this topic is taught in agriculture, it is important to look at from a mathematical point of view because it does involve math.  The practical uses are obvious if you are a farmer but it is also a way of making money in today's world of finances.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If the 8 inch American Bullfrog can jump 83 inches in one hop, how many jumps will it take the Bullfrog to cover one mile?

Warm-up

The 3 inch South African sharp nosed frog can jump 130 inches in one hop.  How many feet is that?

Visualizing Prime and Composite Numbers.

 I know in high school we always define a prime number as having only two factors, one and itself while composite has at least three factors but visually how does that play out?  How do we make pictures that show that idea.

I read something in the "Joy of X" by Steven Strogatz that flipped a switch in my brain to give me a pictorial way of understanding the difference.  I'll go through the explanation step by step here with the appropriate pictures.  I think it makes more sense to students than using just the standard explanation.

Lets start with the numbers one, two, and three:

If you look at the picture there is only one way you can arrange the blocks so you have a rectangular shape.  You cannot rearrange the blocks to change the measurements.  The only way these can be arranged is so it is 1 by either 1, 2, or 3.

Yes, students will say for two that it is  1 x 2 or 2 x 1 so they are different but you can show that the 1 x 2 is rotated to become 2 x1 which brings transformations into visualizing prime and composite rather than applying transformations only to geometry.


Now for the number four.

Four can be drawn as 4 x 1 or 1 x 4 rectangle but it can also be drawn as a 2 x 2 square.  This shows that four times one and  two times two both equal four.

This means that four is composite because there is more than one way to represent the number four .  This shows all the factors for four visually and makes it much easier to understand.



Lets look at five.
When you draw five in the normal rectangular arrangement, it works but if you try to rearrange it into any other configuration, it won't give you a rectangle or square.  It gives you something that is missing part of the shape.

This indicates five is a prime because the only arrangement you can do is a five by one, nothing else gives a full rectangle or square.  Any other arrangement, and something will be missing.  That missing part is what indicates it is a prime.

Furthermore, the five by one shape represents the 5 times 1 multiplication that works.

Finally six.

If you look at the picture, I was able to represent six in two different ways.  The first is in the six by one configuration which is the standard way of showing all numbers but it is the three by two arrangement that makes it a composite number.

So basically, prime numbers can only be arranged in one configuration while composite numbers can be arranged in two or more shapes.  The larger the number, the more arrangements are possible.

If you look at 100, you should be able to draw a 1 by 100 rectangle, a 2 by 50 rectangle, a 4 by 25 rectangle, a 5 by 20 and a 10 by 10 rectangle so it is definitely a composite.

I've found this explanation is easier for students to understand the difference between prime and composite.  The pictures make turn an abstract definition into something more concrete.

I admit that until I read the explanation in the book, I'd always used the verbal definition.  I'd never seen any visual explanation because I'd never connected the number as representing area and the factors represented the lengths of the sides of either a square or rectangle.

Let me now what you think, I'd love to hear.  Have a great day.

Multiplication Games

  I realize I was going to talk about providing visuals for certain algebraic concepts but I am taking time to look at multiplication games because there is a 20 minute time period written into the regular schedule for interventions.  Most of the lowest performing students do not have their multiplication facts down cold.

I realize they can use calculators but there are still some tests to get into certain vocational programs out there that do not allow the use of a calculator.  Furthermore, if students know their multiplication facts then they can use their brain for more difficult things.

These do not require technology to do.  I live in a place where the power can easily go down and there is no access to the digital content.  I like having things like this available, just in case.

1.  Back to Back - the teacher chooses two students to stand in front of the class, by the board and they stand back to back.  When the teacher says go, each student writes down a number between 1 and 20 but they can't see each other's numbers. Someone from the audience multiplies the two numbers together and calls out the product.  The two people at the front of the room take turns guessing the other person's number based on knowing their number and the product.  The first person with the correct answer is the winner and stays up there.  The other person sits down and is replaced by another student.

2. True or False - Divide the students into small groups.  Give each group a double sided index card with true on one side and false on the other side.  The teacher writes a mathematical statement on the board or project it on the white board that is either true or false.  Let the students discuss it for a minute or two and then ask them if they think it is true or false by holding up the index card.  Every team gets a point for their correct answer.  The team that wins is the team that reaches 10 points first.

3.  Beach Ball Toss - Inflate a beach ball and write multiplication problems on the ball before class.  Have the students stand in a circle and toss the ball towards the other side.  The student who catches it reads the problem closest to their right pinky finger out loud and answers it.  The student then tosses it to someone else who repeats the whole thing.

4. Multiplication Scoot - Set up a series of question cards around the room creating different stations. Divide the students into a number of small groups no more than the number of stations.  Give each group an answer sheet and have each group go to a different station to begin the game.  Each group must answer the questions on the card at the station by matching the letter on the card with the letter on the answer sheet.  After a few minutes, tell students to scoot and they move to the next station.  Repeat until they've visited all the stations.  At the end, students turn in their answer sheets and the group with the most correct answers are the winners.

5.  Bingo.  Pass out blank bingo cards.  Let students fill in their choice of answers from the multiplication table.  Make sure the multiplication tables are put away.  Pull the multiplication problems from a container such as 5 x4 or 8 x 7. Students will mark out any products they have that match the problem.  Who ever gets a row covered first or total black out wins.

6.  Check the internet for a jeopardy game on multiplication.  I've always adjusted the rules so all the students who have the correct answer get credit, not just the first one with the correct answer.  To do this, I have students write their answers on a white board and show the answer when I ask for it.  They have to include "What is" as part of the answers.  The group with the most points at the end are the winners.

7. Create a set of multiplication index cards so one side has a problem and the other side has an answer but not the answer to the problem.  It should be written and read as "I have 5 x 4 =_____."  The student fills in the answer of 20 and then says who has 20?  Any one with the answer of 20 can go next.  They read the problem on the front of the cards and gives the answer and asks who has the answer.  This person is out and cannot participate until the game is over.  If you do it properly, everyone has a chance to do it once.

There are more games out there and I'll add a few more ideas another time.  Let me know what you think, I'd love to hear.  Have a great day.

Trying To Get Ready For Next Year

 Today, the high school will be meeting to figure out the whole schedule and populate classes.  We do this knowing everything could change over the summer should the state Department of Education decide not to start on time, or split everything into smaller classes.

We decided to go ahead because we want everything to be as normal as possible.  Since this is a spring ritual, we are doing it as if everything will start on time with regular classes.

There is one big difference in planning for next year.  The principal decided the science and history teachers will be teaching one eighth grade class each which means one less high school class for each.  Unfortunately, they noted the way the schedule is set right now, there is a conflict between a required math class and a required science class for the same group of students.

No we do not use any computer program to get it all done.  We still operate with a white board, marker, and lots of chocolate because things will erased a lot, rewritten, and readjusted.  I already know I'll be teaching one Algebra I, two Algebra II, and one pre-calc.  This school switches Geometry and Algebra II every year to make it easier for them.  I will also be teaching one elective, probably Academic Decathlon.

I expect it will take most of the day.  Since I've gotten to know student's personality, I'll be separating certain students so they group better in the Algebra II classes.  I realize that most school let the counselor run it through a computer program but out here, we work with the counselor to get the classes right.

We know, we might have to go back and redo it depending on what the state requires for the next school year.  At the time of this writing, we still have under 400 infected with only 10 deaths but things are changing as the number of people come up to fish and test positive.  We watch the numbers closely.

We will have figured out how to split students into morning and even sections should it be necessary, or what to do to make sure students stay safe come fall.  We will be ready, let me now what you think.  On Wednesday, I'll address making visual representations for some of the algebraic concepts.  Have a great day.

Warmup

If there are on average 18 strawberries in a pound, how many pounds do you have if you have 1852 strawberries?

Warmup

If a redwood grows eight feet per year and it is 120 feet tall, how old is the redwood tree?

The Math of GPS

 GPS sort of works in Alaska.  If you use the part that loves to give directions from point A to point B, it doesn't always work.  It will have no record of certain roads and show roads as being there that aren't.  On the other hand, the GPS can be used to identify the International Date Line.

Yes, I was in a place where they used GPS to find the International Date line so we could step across it into Russian territory and back.  A GPWS works by triangulating it's position based on distance from satellites.

The GPS measures the distance from a satellite by how long it takes for the signal to be received while keeping track of the location of one, two, or three satellites circling the earth.

One way is to use distance formulas similar to the one we use when calculating the distance between two points.  Instead of using x and y coordinates, these formulas include the z coordinates and the speed of light times the off set time of the receiver..  So basically it is d = sqrt((x-x1) + (y - y1) + (z - z1)) + c1 where c1 is the product of the speed of light times the offset of the receiver and is calculated  for each satellite in the area to create a system of equations.

 The equations in these systems are multivariate and non-linear.  One method used to solve this is Newton's method which requires one to make a guess do a bit of calculation, adjust and continue till you get the answer. Another method for finding distance is to measure the phase difference between incoming and out going continuous waves which uses basic trigonometry to find this.

The above is a short explanation, but much of the math is beyond what most of my students can do so this site has a great 27 page booklet filled with some great math activities to use in class.  It begins with having students do some real calculations.  Then it looks at the Pythagorean theorem's application to the GPS.

After going through the explanation of how GPS works, it has a variety of A variety of activities including Geocaching, using a GPS with a Topographic map, and an activity from Dr Math that uses spheres in 3D.  There are also two to three pages of links that can easily be used in the class.

This site has a nice activity to help students learn to find a position using GPS.  It is a bit simplified and uses the rate x time = distance formula but it isn't bad.  The activity is designed for students to work in groups of 3 to 4 people and is quite clear.

Both of these activities gives students the opportunity to how math is used to help them find their location.  Let me know what you think, I'd love to hear.  Have a great day.

Cooking Ratios Part 2

 This is the continuation of the ratios I began this past Monday.  Although we often discuss ratios and mention they are used in cooking, we don't usually provide details.  So I'm doing that here.

9.  Biscuits - use three parts flour, two parts liquid and one part fat to create the best biscuit.  My father usually used those pop and bake biscuits to go with gravy for breakfast.  I usually prefer eating biscuits with stew or soup.

10.  Crepes - those thin pancakes from France have a ratio of one half part flour, to one part liquid, to one part egg.  For this mixture, it is best to let it sit for a couple hours or over night.

12. Muffins or quick breads use the same ratio of two parts flour, two parts liquid, one part egg, and one part fat makes up the basic structure.  Baking powder is added to help it rise along with sugar to sweeten it.

13. Vinaigrette - the vinegar based dressing is always going to use three parts oil to one part vinegar.  The choice of oil, vinegar, and herbs changes the flavor of the Vinaigrette.

14. Brines - the solution used to make corned beef, or soak chicken in is based on twenty parts water and one part salt.  Cooks then add sugar, and spices to create the unique taste of the brine.

15. Stock - the ratio of parts water to one part bones based on weight is the starting point for any good cooking stock.  Use this for left over chicken, turkey, pork, or beef bones.

16. Bread - is based on five parts of flour to three parts liquid with one teaspoon of yeast per pound of flour.  If a person uses eggs in their bread, those are considered as part of the liquid.  The flour must be weighed out because the volume varies from type to type.

17. Homemade Pasta - is based on three parts flour to two parts egg.  There is usually no other liquid added to the mixture.

18.   Coffee -the ratio depends on the method used to make the coffee.  If you use the drip method of brewing, it requires one part coffee to 17 parts of water.  On the other hand, if a person uses a French press, it works out as one part coffee to 14 or 15 parts of water.

19.  Rice - this depends on the type of rice.  In general if you cook white rice it is one part rice to two parts water while brown rice uses a ratio of one part rice to three parts water.

20.  A simple sugar syrup is made from one part sugar to one part water.

There are so many more ratios associated with cooking out there but these 20 are the basic ones.  We spend so much talking about ratios in class but we usually choose examples that compare boys to girls, or chickens to turkeys or something else but these are real life ratios that are used by professional chefs and cooks.  If you have any budding cooks in your class, they will really love this.

Perhaps you can have students compare various recipes to the actual ratios.  Perhaps you can grab a couple scales from the science department and actually weigh many of the basics before measuring them in cups to see how they compare with the recipes.  Let me know what you think, I'd love to hear.  Have a great day.

Baking Ratios Bakers Need To Know. Part 1.

 The other day, I read something from Food Network that listed ratios all good bakers should know by heart.  I sometimes do ratios in Algebra I so I'm wondering if I can borrow the home economics room to have a cooking lesson using these ratios.

These ratios have been worked out to provide the best product possible.  For biscuits, following the basic ratio will give you the flakiest biscuit, the ones that make your mouth water.

Honestly, I'd heard about a ratio for pie crusts but not for anything else.  My mother was the type who couldn't really cook so she would use boxed mixes as I grew up.  I learned to cook from scratch out of desperation but it has only been recently, I've learned about these ratios.  I plan to put them on an index card to place on my refrigerator.

One thing  before I list the ratios.  These ratios are set up by weight, not by volume so weigh everything on a scale to have everything work out.  Yes, I realize this is a math column but cooking is a combination of math and chemistry.

1.  Pound Cake - the one item I love to use as my base for Strawberry Shortcake.  I prefer the pound cake to a biscuit base since it seems to soak up the juices better.  The basic ratio is one part flour to one part egg to one part fat to one part sugar.  Make sure to take the fat usually in the form of butter and eggs out about two hours before you begin baking so the mixture creams better.

2.  Pancakes - requires two parts flour, two parts liquid, one part eggs and half a part fat to get nice fluffy ones.  I admit, I usually take a good pancake mix and turn them into waffles because I prefer nice thick waffles over pancakes.

3. Meringue - in this case the recipe ratios are for the cookies made out of this mixture.  You will need to use Two parts sugar, one part egg whites, or one part sugar and one part yolks for the best results. One thing is to make sure you do not open the door while the meringues finish baking in the cooling oven.

4. Pate Choux - is the dough used to make eclairs or cream puffs.  This dough requires one part flour, two parts liquid, two parts eggs, and one part fat.  One mixes the liquid, fat, and eggs first, bringing it to a boil over medium heat before adding in the flour and stirring until the mixture pulls away from the sides.

5.  Pie Crust - uses three parts flour to two parts fat to one part liquid.  Always cut the fat into the flour before adding the liquid and do not mix it very much.  If you par bake the crust before adding the filling, the pie does not get the soggy bottom we sometimes experience.

6.  Fritter - the apple or corn fritter that is fried in oil.  The fritter dough uses two parts flour, two parts liquid and one part egg.  This makes the great basic dough but when frying them, it is best not to have too many in the pan.

7.  Cookies - this is the basic ratio often associated with sugar cookies.  You need to have three parts flour, two parts fat and one part sugar for the dry ingredients.  The number of eggs vary according to the type of cookie.

8.  Custard - the custard made from eggs and baked, not from a powder.  To get the best custard, one needs to use two parts eggs to one part liquid.  There is usually sugar added but just enough to make it sweet without being extra sweet.

I'll be sharing more ratios in Wednesday's column.  Maybe some of the students would enjoy trying some of these out at home.

Warm-up

If 150 avocados weigh 60 pounds, how much does one avocado weigh?

Warm-up

If it takes 18 pounds of apples to make 1 gallon of juice, how many gallons will you get if you have 225 pounds of apples?

Cost of Cars from 1950 On

  Since most schools still have several weeks left before the end of the year, teachers are still having to come up with activities that allow students to do math while keeping the assignment interesting.

I stumbled across the fact that the Model T Ford sold for $850 back in 1908 which is around $21,000 when adjusted for inflation.  If you waited till 1916 to buy the Model T, the price dropped to $360 or around $7,000.

That got me to thinking about how the cost has changed over time but if the same prices were adjusted for inflation, have costs changed all that much?

If you look at this  site lists the average cost of vehicles in the years from 1967 to 2016.  The article gives the price of a car for that year it was released, the adjusted price in 2016 dollars and the increase or decrease in dollars making it possible to calculate the percent increase or decrease using 2016 dollars.  The average costs means it is not for a specific model, just for what you might spend.

On the other hand, this site has some great information on the cost of both new and used vehicles beginning in 1950.  The brand new Kaiser-Frazer Henry J cost just under 15,000 at the time while a used 1947 4 door Studebaker ran almost $12,000. If you couldn't afford that, you could have purchased a 1936 2 door Ford Model 48 for under $3000.  This site covers specific cars from 1950 to 2020 and usually offers more than one new car and more than one used car so people can see how they compared.

In addition, the prices listed adjusted for inflation and shown as the price might be in today's dollars to make the comparisons much easier.  If students want to know what the price was say in 1950, they can use this calculator to figure it out.  All they have to do is put in the amount, 2020 for the first year and the other year such as 1950 and it will tell you how much it was worth in 1950.

Both sites have enough information to create graphs showing increases in the cost of new cars either in dollars or percent increase over a specific time period.  Students could also create an explanation of the increases and include thoughts of the various cars.

This site  looks at how much a $15,000 actually bought beginning in 1953 through 2020.  The article has the actual values, a graph showing the percent increase or decrease each year.  This site is different than others because it takes time to explain how they calculated inflation rates and it gives students the opportunity to practice reading and interpreting data from graphs and tables.

Another way to look at cars is looking at the number of cars sold each year between 1951 and 2016.  The original graph is a line graph but students can take time to read and interpret the data to translate it into raw data.  The graph is such that students can calculate the percent increase or decrease from year to year.

So many different ways to present and interpret both graphs and tables.  This can be turned into a project easily done at home using both spread sheets and word documents.  Giving students the opportunity to interpret data and graphs, or move from one to the other and then create a word document to discuss it.

I honestly feel as if students do not get enough practice with graphs, either making them or reading them.  These possibilities provide the ability to both make and interpret graphs.  Let me know what you think, I'd love to hear.  Have a great day.

From Thinking To Understanding Via Headlines

 In math webinars and trainings, I keep hearing that both we and the students must use visible thinking.  Of course the first thing that always pops up in my mind is the techniques used during reading.

When I taught reading, I had to make my "thinking visible" to my students.  In other words, I had to "think" out loud so they could see how I figured out the information.  That isn't always how we do it in math.

I've heard it said that understanding is the result of thinking and not a type of thinking at all.  Based on that, students have to use thinking to get to understanding.  It is also said that students are most engaged when they are involved with minds on thinking.

It is also said that learning is a direct result of thinking rather than memorizing steps or mathematical facts.  It is possible to solve equations while following algorithms but that does not guarantee understanding or learning. So let's look at some ways to help students develop from thinking to understanding.

1.  One way to do this is by using headlines which can be used in multiple ways.  This exercise uses the idea that students can capture an idea, concept, or topic in a few words.  Headlines are written with only enough essential information to give the reader an idea of what is being shared in the article  while capturing the attention of the reader.

To use it in the math class, you might have students write a headline which summarizes the most important idea on a particular topic at the beginning of class.  An example might be "Pythagorus quoted as saying "a^2 + b^2 = c^2Then later on, at the end of class or perhaps the next day, ask the students how they might change the headline after the lesson before having them explain how the headline differed.

A good way to implement this use of headlines is to have students think about the big ideas or important topic for the section.  Then let them brainstorm in groups as to what the big idea or central theme to help clarify thing.  Then in pairs, they create the headline.  Once everyone is done, they can share the headlines as a way of promoting understanding.

Two advantages to having students use this activity is that it provides a quick assessment of understanding and it means the students have to focus on the concept rather than using catchy phrases. In addition, this activity lays the foundation for students to be able to connect to other future topics.


Another way to use it is to create a short headline type story without the question.  Have the students read the story and then create possible mathematical questions that could be answered using the information.  An example might be:

1. "Worldwide, about 10 million metric tons of olives are produced each year.  A million metric tons are used for table olives and nine million (93 percent of the total crop are pressed for olive oil."

What mathematical questions could you ask that can be answered by the information in the headline?

Another way to use the headline story is to include the question but have too much or not enough information so students have to figure out what information can be used to answer the question if there is too much or what information is needed if there is not enough information.  If the story has too much information, have students pose other questions that can be answered with the information. If the story does not have enough information, ask students to create questions that can be answered with the information given.

This can easily be done in class or via distance.  Let me know what you think, I'd love to hear, have a great day.

Comparing Toyo Vented Heaters

Most Alaskans do not have a furnace in the basement or a wall furnace.  We have vented indoor furnaces which are used to heat our houses.  They are vented outside and have connections to outdoor fuel tank.  The last exercise I designed for my students to work on compares three models of Toys vented furnaces.  I'd hoped to include another brand but they went out of business a few years ago and left only one.  

Today, we are looking at three different Toyo stoves.  You are looking to replace your current stove with a new one.  The information is given below. 

You will need to do the following calculations.

1. Find the volume using length x width x height.  Remember 1/2 - .5, 3/4 = .75 and 3/8 = 3.75.

2. Calculate the number of days  250 gallons of heating fuel will last for each stove for high, medium, and low. Do the calculations this way.

      a.  Divide 250 by gallons used per hour = number of hours.

      b.  Divide the number of hours from part a  by 24 to give you the number of days.  Round down.

3. If heating fuel costs $5.25 per gallon, how much do you spend every hour the stove runs on high, medium, and low.  

      a. Multiply the number of gallons per hour times $5.25 to find the answer for high medium and low.

Fill out the following table:

Stove	L-750	L-60	L-560
Volume in^3
Number of days of fuel High
Number of days of fuel Medium
Number of days of fuel  Low
Cost per hour high
Cost per hour medium
Cost per hour low
Weight

Please answer these questions using complete sentences.

1. If the area of your house is 1600 square feet, which stove would you you buy and explain why.  

2.  Your grandmother needs a new stove for her house.  Her house has an area of 1200 square feet.  Which one would you suggest she buy and justify your answer.

Warm-up

If a semi gets 6.5 mpg and the tank holds 250 gallons of gas, how far can the person drive on three tanks of gas?

Warm-up

If this truck is said to average 22mpg, how many gallons will you need to drive 852 miles?

Comparing Electrical Tankless Water Heaters.

 I had one of the staff members ask me about tankless water heaters, so I created a unit on the topic.  I was looking at both electric and liquid propane tankless water heaters but the electrical ones had the most data so I shoe those.  Tankless water heaters are also known as 

Today you are checking out three different “On Demand” or Tankless Water Heaters.  They are all electric, small and fit on the wall.  The size was chosen to meet the needs of family with 6 members at least.

Time to do some calulations.  GPM is gallons per minute while the flow is based on the temperature of the incoming water.  Notice that as the temperature of the water flowing in increases, the more water you get each minute.

1. Find the volume of the water heater. You will multiply height x width x depth.

2.   Find the percent increase for each increment of increase in temperature.  To find it, you will subtract the (new temperature - old temperature) divided by the original temperature and multiply the answer by 100.

Example for the Rheem going from 4.2 at 37 degrees to 5.1 at 47 degrees.

(5.1 - 4.2)/4.2 = .21 x 100 = 21 percent.

Do this for all the GPM for each brand.

3.  What is the percent difference between the highest and the lowest for each temperature.  So for the flow at 37 degrees, the percent difference between the Rheem and Ecosmart.

We use the same formula from step 2.

(4.2 - 3.6)/3.6 = .16666 x 100 = 16.7 percent.

So you’d repeat it for the 47 degree, 57 degrees, and 62 degrees.

Then you’d do the same thing fo the 47 degrees, 57 degrees, and 62 degrees.

Fill out the following table.

Brand	Rheem	Ecosmart	Black & Decker
Volume
Percent increase 37 to 47 degrees
Percent increase 47 to 57 degrees
Percent increase 57 to 62 degrees
Percent diff at 37 deg
Percent diff at 47 deg
Percent diff at 57 deg
Percent diff at 62 deg

4.  So based on all the information printed in the table at the top, which would you recommend your family buy.

I had fun creating these.  My principal suggested I put these exercises in my sub binder for next year, if I'm every suddenly absent.  I hope you like this.  Let me know what you think, I'd love to hear.  On Monday, I'll share the last activity I created for the last part of school.  It will be on furnaces.  Have a great weekend.