Preschool Math (Pre K-2nd Grade) using Math Their Way

 I do this list of activities in this basic order, spiraling around each of these four years. I stop each area when the activities seem to hard for them, because I know I will be getting back to them the following year, or when they seem to have mastered the content of that area.

Weeks 1-5: Free Exploration and Counting
 Introduction
Application and Extension
Counting (Assessment)
Counting On
Learning to Write Numerals


Weeks 6-9 Patterns, part one
 Introduction
 Application and Extension



Weeks 10-13: Sorting and Classifying
Introduction
Application and Extension
Graphing: Introduction


Weeks 14-17: Comparing 
     Introduction
    Application and Extension

Weeks 18-28: Number
(Assessment)
Concept Level
          Introduction
          Application and Extension
Connecting Level
Symbolic Levels

Weeks 29-36: Place Value
(Assessment)
 Introduction: The Counting Game
 Application and Extension


Weeks 37-42: Patterns
Patterns, part two 

source: Math Their Way

Place Value with Decimals

1. Review place value in general. Read How Much is a Million? by David Schwartz. Practice reading and writing large numbers in numeral and written form. David Schwartz tells us that it takes one gallon of water to keep a one-inch goldfish alive. How many gallons of water will it take to keep 10 goldfish alive? 100 goldfish? 10, 000 goldfish? One million goldfish? There are 7 1/2 gallons of water in a cubic foot of water. How many cubic feet of water are necessary to keep 10 goldfish alive? 100 goldfish? 10, 000 goldfish? One million goldfish?
2. Begin demonstrating decimals on a trading board. If you used one of these to introduce addition and subtraction, now you can just shift your child's thinking by telling him the one cup on the trading board is now one, instead of ten, then one bean is 1/10. I told him to think of it as if the cup represented a class at co-op which had 10 students in it. The class would be one, and each student would be 1/10 of the class. If we decided that the bowl represented one (instead of 100 as it had in the past), then we could look at it as one co-op school, in which each class was 1/10 of the co-op (assuming that the co-op had 10 classes in it) and each student was 1/100 of the co-op students. After this introduction, we practiced ways of writing fractions whose denominators are powers of ten. We wrote them as fractions and we wrote them as decimals.
3. Next, practice the same concept using different materials. Assign fractional values using plastic colored chips on the counting/trading board. Together create decimal problems for addition and subtraction using the counting/trading board by rolling a die for each column. Since he already knows how to create addition and subtraction problems in this way, the only difficulty he might encounter is where to place the decimal. 
4. Next get out a piece of 10 x 10 graph paper to make a matrix. You can use Math-U- See blocks, Unifix cube sticks, bean sticks or whatever you have as a base-10 manipulative, since we are working with decimals, which are all working with divisions of ten. Use whatever you had used to represent 10 and call it one. Using a series of questions, you can guide your student to see the relationships.
5. Have the student read decimal fractions through ten thousandths and decimal fractions with whole numbers. Have your student write the following rules in his math journal: 1.) The decimal point is read as "and." 2.) The first decimal place is tenths, the second decimal place is hundredths, the third decimal place is thousandths and the fourth is ten thousandths. 3,) Read the decimal number as a whole number followed by the name of the ending place.
6. Play Decimal Place Value with Playing Cards. Instructions and a printable score keeping sheet can be found at Games for Gains. Basically, you have a deck of playing cards without the face cards, and in this game the 10 card represents 0. Each player takes a card from the pile and decides which place value to assign the number on the "Round 1" portion of their score sheet. The player can only write that number under one place value column (hundreds, tens, ones, tenths, hundredths, or thousandths). Once each player has written down the number in a place value column, it cannot be changed at any point during the game. This is repeated until all 6 columns are filled. The winner of that round is whoever has the largest number.
7. Connect tenths and hundredths to dimes and pennies. Questions like, which is larger, 1.8 or 1.09? may suddenly seem easier if translated into money terms, $1.80 or $1.09.
8. Connect decimal numbers meterstick, since it is already divided into hundredths.

Resources: 

• How Much is a Million? by David Schwartz
• Mathematics; a Way of Thinking, Bob Baratta-Lorton
• The Math Maniac
• Games for Gains

Make Your Own Medieval Shoppe (and Practice Math Skills)

Day 1: Research Research what kinds of shops were available during the Middle Ages and then pick from them what kind of shop you want to make. You will start with 10,000 gold pieces. In this set-up 10 copper pieces equals 1 silver piece and 10 silver pieces equals 1 gold piece.

Day 2: A Sign To ensure that customers will know what you sell and be able to find you, you will need a sign in front of your shop. The sign maker has four sizes to choose from: the smallest will cost 25 gold pieces, the next size up will cost 50 gold pieces, one a little larger still will cost 75 gold pieces and the largest sign will cost 100 gold pieces. Which do you want to buy? You will have opportunities to buy larger ones later.

Day 3 Inventory Make up your inventory. What things do you want to sell? Decide on a price that you will purchase them for (generally it at least half the price you would expect to pay for the item if you were buying it.) The prices do not have to be historically accurate, but they have to make sense to you. You can't have a full suit of armor cost less than a knife, for example. Decide on about 25-35 items, and have a wide variety of prices, ranging from 50 gold pieces down to 7 silver pieces, with a variety in between, so that all 25 or so items equal about 240 gold pieces.
Day 4 Stocking the Shelves Now that you have your list of items available to you to buy, how many of each will you buy? Keep these things in mind as you decide on your inventory:

• You don't have to have all of them. 
• You do not want to miss a sale by not having the item in stock.
• You will need to have money to pay your bills.
• You will be able to buy more items at regular intervals as you go along.

Day 5 Determining the Costs Once you have decided how many of each item you want, multiply the costs you determined on Day One by the quantity you determined on Day Three. Add all of these up and subtract it from your start-up money (10,000 gold pieces.) You must have some left over, remember, for your expenses.

Day 6 Determining Prices Decide on your selling prices. Most stores mark up the prices between 100 and 150 percent of the price they bought the items for. Determine both 100 and 150 percent of the cost to you for each item, and that will determine the range of your selling price. Remember in the Middle Ages, people were much more likely to haggle prices than we are today, so you need to have a range for the prices. You want to offer the item for the highest price first, but then you may be haggled down to your lowest price before you can actually sell the item! Write down the range of prices next to each item.

Day 7 Preparing for Tithing You need to pay the church a percentage of your earnings so you have to make sure you will have this amount leftover at the end. You must pass this on to your customers. Ask your parent whether the church will be expecting a 5, 6 or 10% tithe and add this to each of your prices. (Mom- use the percentage that you want your student to work on or is in keeping with his skill level.)

Day 8 Bookkeeping You need to keep track of all the items in your store, what you have sold and what you need to order. Fill out a bookkeeping sheet as you go along. 

Day 9 Debt and Credit You will also need to keep track of all your transactions, your money out (debt) and your money in (credits) and keep track of the balance of gold pieces you have left.

Day 10 Fill Orders Now your friends and family can give you orders for you to fill. Have them pretend that they are people in the Middle Ages and have them pick from your inventory.
Example:
Mr. Baker wants 2 sets of leather boots, 1 steel shield, 2 short bows and 1 long bow.
Mr Carter wants 1 steel breastplate, 1 pair of steel boots, 1 Great sword, 1 long sword and 1 wooden shield.
Figure out how much each of your customers owes you.

Day 11 Pay Rent The Lord is collecting his quarterly rent. You must pay him 200 gold pieces ,40 silver pieces and 6 copper pieces (or $240.60).

Day 12 Additional In-store Sales Ask your parents to give you additional sales. (Mom -This is where you get to include some of the math problems from their texts or that you would otherwise assign them. If your student picked the smallest sign on day 2, give them one in store sale, the next size up will attract two in-store sales, the third size will incur four in-store sales and the largest sign will award your student five in-store sales.

Day 13 Fortune Life has its ups and downs. You will need to make these up (or for more surprise, you can get a parent to make these.)

Day 14 Pay your tithes Add together all the tithes you have collected and pay them to your bishop.

Day 15 Order Inventory Now it is time to replace the items you sold this month. Write down the quantities, multiply them times the cost, add all the extended costs to get your total. Write your new amounts on your inventory sheet.

Day 16 Ledger Look at all your money in and out and keep track of them on a Ledger Sheet.

Day 17 Profit or Loss? Once you know the total amount of debits and credits, determine which is higher. Did you have a profit or loss? By how much. Subtract the smaller total from the larger total to find the difference.

Days 18-on You may repeat days 7-16 as many times as you desire. If you do a full 12 mo the worth (each time you run through it is one month), you can determine your year's profit or loss by adding together/subtracting each month's profit and loss to see how you did running a Medieval Shoppe.

Triangular Numbers, More on Pascal's Triangle and Odd/Even Numbers

Remember when we played with squares and square numbers?

I thought maybe it was time to look at triangular numbers.

We took out circles because triangular number is the number of dots in an equilateral triangle uniformly filled with dots. 

Okay, one is the smallest way to make a triangle. How many circles does it take to make the next size larger triangle? We played and built.

Three.

Next size up?

Six.

Can you predict the next size up triangle?

Nine? Is it multiples of three?

Let's see.

No, ten.

Let's build one more.

Fifteen.

Now let's take them and make squares out of them. How many more do you need?

One more to the three.

Three more to the six.

Can you predict the next amount it will take to make the triangles a square?

Six?

Let's try and see.

Why did you say six?

Because it was the same sequence of numbers...1...3...and 6.

The sum of two consecutive triangular numbers is a square number.

Then I told them a bit about Karl Gauss. He lived in the 1800's and he proved that you can make any whole number by adding no more than three triangular numbers and that triangular numbers never end in 2, 4, 7 or 9.

What triangular numbers add up to your age.

Ten is already a triangular number but it can also be made by1 +3+6!

Seven is 6+1!

Thirteen can be made by adding 10 + 3.

What about you? Hmmmm...49 can be made by adding 28+28+3.

Lastly, remember Pascal's Triangle?

Can you find triangular numbers in it?

Then I found this investigation of number patterns and Pascal's Triangle and I had to try it out too.

So, I printed out a copy of Pascal's Triangle and we went to work coloring multiples of two and three.

We tag teamed this activity by starting with Quentin until he wanted to pass the marker to James and then James to Sam.

You may find it helpful to know that the sum of the digits of any multiple of 3 is itself divisible by 3. For example, 252 = 2+5+2 = 9, and 9 is divisible by 3; 924 = 9+2+4 = 15, and 15 is divisible by 3—so both 252 and 924 are colored, but 560 = 5+6 = 11, is not.

You could also look at it in terms of odd and even numbers and if two cells above are of the same type (both odd, both even), you color the cell below even. If the cells above are different (odd + even, or even + odd) you color it odd. Why does this work?

Because of the principle that an odd + an odd always equals an even. Same is true with the sum of two even numbers is itself even. If your student doesn't know this, he will need to play with numbers to prove it to himself. With every step of the way, he needs to see for himself what the rules of math are, and not just take it from you. 

What To Do If Your Student Is Not At Grade Level in Math- A Learn Math Fast System Review

 

What can you do if you discover your 4th grader is barely passing 2nd grade math or your 6th grader is all over the place with his math skills -in some areas he is advanced but others is his behind? You see high school looming and you cannot figure out how your student is going to possibly make up years missing math concepts by next year. These are not unusual places to be, especially if your student has been learning by a hands-on or living math method or if your student has learning disabilities or is gifted.

I know the first instinct is to make your student spend more time with the math books or drill the concepts, hoping that the skills will begin to come. This often leads the struggling student to hate math, and the battles begin. I would like to suggest an alternative.

Don't push your struggling math student to hate math.

If you have been following my blog for very long, you know that I have heartily endorsed a hands-on and living books style of math instruction, and I still do like it for a foundation in math concepts. But what do you do if you have completed Math Their Way, Mathematics, A Way of Thinking and Living Math and your student still is lacking the necessary math skills? If you have been in my shoes then you might have experienced the difficulty of presenting a math book at their level that is obviously for younger students because of the pictures and the subject matter discussed.

What do you do when you feel like your child needs to start over from the beginning?   

We faced this problem this year and I am happy to say that I have found the solution. I found The Learn Math Fast System and I was so impressed by what I found on the website that I, uncharacteristically of me, emailed the author and asked if I could have a sample to review. I was sent the whole set of four volumes and a geometry kit. When materials were sent, I was told by the author that the program is designed to be read from page one, which starts with first grade math, all the way through to the end of book four, getting your child caught up to eighth grade math in about a year. I must admit that I was a bit skeptical that my struggling students could catch up that fast.

But we began to get moments in which the math concepts that they had been struggling with for so long became clear for the first time. Quentin had his first "a-ha" moment when we were going over fractions. The program approaches each topic in many different ways so even though in the beginning he had that confused look, his face suddenly cleared and he exclaimed, "Now I understand!" He answered all the remaining questions correctly.

There are no manipulatives, so it is not hands-on. The graphics in the book are fantastic, however, and they are simple and straight-forward, nothing cute or dumbed-down. This gives it a hands-on feel to it, and makes the more simple concepts not seem too babyish. It is written in a tone that is lively and engaging, which helps to take the fear out of math. The program gives lots of mental math tips, practice and timed tests (although we have opted out of the timed tests due to learning disabilities and anxiety.) The math facts are taught using a systematic approach to ensure that all gaps are filled in, giving students a solid foundation and understanding of math, preparing them for high school math. Part of the reason that the system works so fast is that it focuses just on math concepts. For this reason, there are a few topics that you won't find in these books. For example, learning to tell time, Roman numerals, tally marks, and the days of the week/year are not included in the Learn Math Fast System.

It can successfully prepare your child for high school math in about a year.

This approach has worked well with both my struggling 4th grader and my 6th grader who was advanced in some areas and less so in other areas. My 4th grader has advanced to grade level material in just a few short months and my 6th grader is now ready for the third book, Pre-Algebra. I believe it would work well, also with an advanced math student because it leaves out all the fluff and the student can gain the math skills he needs quickly and efficiently, satisfying the advanced learner's desire to achieve. 

The program is very inexpensive. The first four books can be purchased for $149 (with free shipping), which is much less than most programs materials for first to eighth grade math. I am sure we will be purchasing Volumes V and VI, which cover Algebra I and II, which cost $49 each (and $10 shipping).

What is inside each book, sample pages and a placement test can all be found at the Learn Math Fast System website. So whether you have a struggling math student, an advanced student or a student with a mixed level of skills, go over and take a look, and see what you think. It is the best no-nonsense math program I have ever seen.

Disclaimer: I received this product free through the company owner in exchange for my candid review. A positive review was not required, nor did I receive any further compensation. All opinions expressed are mine. I am disclosing this in accordance with the FTC regulations.

Top 5 Math Posts of 2013

Money Matters: Tostada Cost Analysis

Spy Mission #1: Mr. Victim's Missing Box

Unstandardized Measurement (grades 3-6) 60

Graphing (Pre-K-2) 101

Coordinate Graphing (grades 3-6) 110

Thanksgiving Week Activity: Turkey Glyph

Today when I announced to my boys that we were only going to do Thanksgiving related activities this week, I don't think they had a math activity in mind! This is another activity from Mathwire.

You have all seen the paper turkeys with multi-colored tail feathers.

Well, the spin on these is that each color represents something on our dinner menu and they had to pick the colors for their turkey's tail feathers that match up to the foods they like and plan to eat from our Thanksgiving menu.
Light Brown- Stuffing
Black- Gravy
Red- Cranberry/Orange Relish
White-Mashed Potatoes
Yellow-Corn on the Cob
Green-Pickles & Olives tray
Light Blue-Deviled Eggs
Orange-Sweet Potato Casserole
Orange Striped-Pumpkin/Cranberry Bread
Tan-Turkey (shaped) Bread
Pink-Tortellini Salad
Dark Blue-Jello
Dark Brown-Pumpkin Pie
Purple-Cake

For the turkey body, you could choose light brown for white meat and dark brown for dark meat.
James wanted both, so we made his turkey body with both light and dark brown.

Quentin's.



Sam's.
Even though this is an early elementry activity, I had everyone in the family do it, so we could have more to graph.



James'

Alex's

Mom's

Katie's

And our vegetarian Dad just made a clutch of feathers.

You can use these to graph preferences.

On a practical level, it can help you to plan how much of each item on the menu to make. And lastly, serve as cute decorations for Thanksgiving Day.
Just as James said, "Each of the turkeys tells a story."

Negative Numbers (3-6)

. 

Postman Games: Addition of Signed Numbers

This week we worked on addition of signed numbers, and began with problems to work out centering around a postman named Sam. Sam is a strange postman because sometimes after he has delivered the mail, he comes back and takes it away again. Sam only delivers two types of mail: checks and bills. The bills are when you owe someone money and checks are when you get money. I showed him how to write positive and negative numbers, and then we went over the cards I had made up on index cards cut in half about whether you were richer or poorer if you got each card. Then I began giving him two cards and he had to write them down as problems and solve them. For example, if Postman Sam gave you a check for $4 and then another check for $3, are you richer or poorer, and by how much. I gave him a grid to put the first problem in, and after that he could refer to it as needed. We then went over the four possibilities for addition of signed numbers in Postman Stories:

1. The mail carrier brings a check for ___, then he brings another check for ____.
2. The mail carrier brings a bill for ____ , then he brings another bill for ____.
3. The mail carrier brings a check for ___, then he brings a bill for____.
4. The mail carrier brings a bill for ____, then he brings a check for ____.

We also went over how they are written.

Postman Games: Subtraction of Signed Numbers

Remember how we talked about the fact that Sam, the postman is not too good at delivering the mail, and sometimes he comes back and takes the mail away again. We used cards again to play a game, this time using subtraction of positive and negative numbers. He wrote them down again in his math journal, paying particular attention to the signs. The four possibilities were:

1. The mail carrier brings a check for ___, and takes away a check for ___.
2. The mail carrier brings a check for ___, and takes away a bill for___.
3. The mail carrier brings a bill for ___, and takes away a check for ___. 
4. The mail carrier brings a bill for ___, and takes away a bill for ___.

Coordinate Graphing and Negative Numbers
Negative numbers are useful when constructing coordinate graphs. They allow students to project their linear graphs as far as they wish, and then speculate on the meaning of all points on the line. 

We used the cards again, to make up addition problems to graph on a coordinate graph.
For, example, if he pulled a -4 card, he would then plug it into this formula:
-4 + X = 10
and then plot that on a coordinate graph to find its solution.

"Mathematics is more than performing basic arithmetic operations, but is a way of thinking. Negative numbers are a tool they can use to expand their thinking." -Bob Baratta-Lorton

sources:

•  Mathematics; a Way of Thinking, Bob Baratta-Lorton
• Discovery in Mathematics by Robert Davis

Sorting and Classifying (3-6)

The goal of sorting and classifying skills is to clarify thinking and improve their facility with language. Forming categories and dealing with their relationships is a part of logic. Sorting and classifying involve the recognition of attributes properties of a given object or group of objects  person chooses to isolate and observe. Intellectural development consists in good part of learning to invent or discover attributes or categories relevant to a particular subject, then dealing with the relationships among various attributes. Through sorting and classifying the student can make better sense out of a wide variety of impressions and handle increasingly complex situations that might otherwise prove overwhelming. They can restrict information by choosing from among attributes, thus limiting the scope of any one problem to something that is manageable.
Buttons are great to start with because you can by a ton of them fairly cheaply and they have clear attributes to sort them by: color, size, shape, number of holes and as to whether they have shank or not.
You can then move on to any sort of object that is around the house; toys or natural found objects.
Once you feel sure that they are comfortable with sorting, you can move on to sorting trees.
Going back to the most familiar is the best way to begin using sorting tree, using buttons.
The tree begins with the first division, big and little. Each of these groups, then can be divided further into round or square under each of the size categories. Then the groups can be further divided by how many holes they have in them, under each of the previous categories. All these divisions form a tree-like formation. 

source: Mathematics; a Way of Thinking, Bob Baratta-Lorton

Coordinate Graphing (grades 3-6)

Coordinate graphs use ordered pairs of numbers to designate a single point on a grid that represents two variables at once. The data from any graph students make can be displayed using ordered pairs of numbers on a grid, but not all data is usefully displayed in such a manner. Only data that can be used to  make specific predictions for future events is best represented and interpreted in coordinate graphs. Why this is so will can be made clear to students. First they need to learn the basic techniques of assembling coordinate graphs. Once these techniques are learned, they can be guided through experiences that help them learn when to use coordinate graphs and when another form of graphing is more appropriate.

Number Pairs

I originally saw this activity done by using a "Number Machine," which is a half-gallon milk carton covered in paper and with two slots in it. An specially prepared index card, with a number on it, is inserted into the top slot of the milk carton and, by going through the carton, the card flips over, to reveal a different number. All of this is to interestingly introduce students to ordered pairs of functional or interrelated numbers. I am sure this concept is better presented in this way in a classroom setting. I just wrote the pairs of numbers on a sheet of paper and Quentin figured them out as logic puzzles. Then he made up a few for me.

Coordinate Tic-Tac-Toe

The next step in learning coordinate graphing is to learn that ordered pairs of numbers can be used to indicate a single point on a grid. James has played the game Battleship many times before, which gave him some experience with ordered pairs of numbers on a grid, but the coordinate points are in the squares and I want him to practice making the coordinate points where the lines intersect.

As I constructed the board for this game, I taught him some terms and the like. I told him that I was numbering the lines that divide the sections and that they are called axes. I showed him that where the two lines crossed, I wrote a zero. On the part of the axis that started at the zero and went to the right, I wrote numbers on each line crossing the axis. I pointed out that the numbers were on the lines and not in the spaces. On the part of the axis that started at the zero and went to the top, I also wrote numbers on the crossing lines. We played Coordinate Tic-Tac-Toe.  To win a point, one needs four marks in a

row. To make a move, the player writes down a pair of numbers. The first number tells where the player is starting on the box axis (that's this horizontal line) and going up. The second number tells where the player is starting on the triangle axis (that's this vertical line) and going across. The player then marks (either an X or an O) on the graph paper where the lines for the two numbers meet. It was clear right away that he understood the concepts, but we played for awhile just for the pleasure of it.

Graphing Coordinates

The next task to learn is how to plot sets of numbers from the previous activities on coordinate graph paper and how to use the plotted points to predict future numbers that have the same ordered pattern. I first had him make a coordinate graph on the graph paper just as I had done when we played Coordinate Tic-Tac-Toe. I then gave him 5 or 6 ordered number pairs and had him plot them on the coordinate graph. I then had him draw a line through all points he had marked so far. I then had him predict the additional numbered pairs that could be plotted on the graph, using the same pattern. After doing a few of these, he is now able to make up sets of cards for others to plot.

Now, I have to show him that a set of numbers may yield sets of points that zigzag or curve. The purpose of his learning coordinate graphing is so that he can graph actual events and not all graphable events yield linear patterns. Soon he will begin marking points on graphs to record events not as carefully controlled as the number I select for him. The cards are used to assist the students in learning how to plot points on a coordinate graph, and to show that some activities recorded on a coordinate graph yield points all in a line. The third group of numbers keeps him from thinking this is true for all activities.

Scientific Investigations and Coordinate Graphing 

For his first experiences at using his coordinate graphing skills to investigate relationships present in events, I will suggest some scientific investigations. 

Rates of change can be measured. As a ball is dropped from higher and higher points, does the height of its bounce change in a predictable manner? 

Do all balls bounce to the same height dropped from the same place? Is the graph for each separate ball and its bounce a straight line? Or, do some balls bounce less high as they are moved to successively higher points? What bouncing patterns do things other than balls have? 

A study of balls can also lead to the study of circles. Is the diameter of a circle related to its circumference? If the diameter is known, can the circumference be predicted?

 We gathered several circular items from around the house. We measured the circumference with a bit of yarn.

 Then he measured the diameter of the item.

 And we plotted the measurements on a coordinate graph. At some point, he felt confident that there was a relationship between the two measurements and drew a line through the measurements that he had already plotted. This gave him points all along the graph.

After this he could just take one measurement, plot it into the graph and see where the line fell and make a prediction at the second measurement of the pair.

 Do squares and rectangles of tiles have predictable perimeters as successive rows or columns of tiles are added? 

How would a graph for the perimeter of a rectangle with a constant number of columns and a growing number of rows compare with a graph for the same starting rectangle with growing columns and constant rows?

Perimeter can lead to area. What happens to the area of a rectangle as more rows or columns are added? Does the same thing happen to areas of other shapes? What happens to the perimeter of a rectangle if its area is kept the same but the tiles are rearranged to make long thin rectangles or short fat ones? What numbers would be graphed to show if there was a pattern? If the graph this question produces is not a straight line, can it still be used to make predictions?

Volume and Capacity

Area leads to volume and capacity. 

First he measured and cut out different sized cylinders from cardboard tubes. Each cylinder of a different height.

Next he was posed with the question of whether or not a relationship exist between the height of a cylinder and the number of tablespoons or cups of rice it will hold?

Could a coordinate graph be used to predict how many cups of rice a cylinder would hold if only its height were known? 

He made a prediction, using the graph that 6 inch cylinder would hold 14 tablespoons, and then confirmed it.

For his own pleasure, he figured out how many tablespoons were in a cup and how this related to each of the measurements. He determined that 1/4 cup equaled 3 tablespoons, so a 1 1/2 inch tube would hold 1/4 cup. He then figured out that since a 2 inch tube holds about 4 1/2 tablespoons and a 4 inch tube holds about 9 tablespoons, then a 6 inch tube would hold about 13 1/2 tablespoons, and he determined that it would be roughly 1 cup.

Capacity can lead to mass or weight. How can a scale give people their weight so quickly? If a rubber band is arranged as shown, can the distance it will stretch as successive numbers of washers are placed on the hook be predicted?

If a graph can predict how far the rubber band will stretch, can it tell how many washers were added? 
Other possibilities for extension: Does it make any difference how thick the rubber band is? Is there a limit to how many washers the rubber band will hold? Can the graph be used to tell when this limit is near? 

Speed is an area of measurement also. Changes in speed, accelerations or decelerations, lead to graphable relationships.One apparatus for measuring changes in speed can be seen through experimentation. 

A toy car is released at the top of the board and allowed to roll freely down the slope. One number indicates the amount of time the car took to make the journey. On the other axis are the inches the board is elevated.
 
After each successfully timed run at one height, the board is raised one level and the car is timed on a new run. I had them graph the relationship between the time the car takes to get down the slope and the height of the board, and they could then predict the future points on the graph. The same apparatus used to measure acceleration can also be used to measure force. The only additional equipment needed is a block of wood at the end of the ramp. Can a graph be made to predict how far the car will push the block of wood as the car is released from successively higher starting points? If a higher starting point causes the car to knock the wood a predictably farther distance, what would happen if a longer board were used or if the block of wood were heavier or lighter? Do the same things cause the wooden block to be hit farther that also caused the car to travel faster in the experiments on acceleration? 

Coordinate graphs are not appropriate in every situation. The students must learn through experience when they are beneficial and when another method of displaying data is more useful. The general rule for distinguishing a topic appropriate to coordinate graphing from topics better represented by another form is to determine if the graph is to be used to make comparisons (other graphs) or to predict

(coordinate graphs). Graphs of any kind are only ways of displaying information; if a student chooses to display data on a coordinate graph and it proves to be of no benefit, he or she may always switch to another form of graphing. Students can learn for themselves when a particular kind of graph is appropriate.

source: Mathematics; a Way of Thinking, Bob Baratta-Lorton