Hands-on Algebra: Polynomials

Just as we used tiles and cubes to learn basic math, we can use them for more advanced math.

For this lesson, you will need to have three sizes of tiles in two colors to represent tiles and their opposites. 

Notice that these are not perfect or measured exactly. How neat and perfect they are does not matter for older students. They get the idea. All you need is two colors of paper and it will just take a few minutes to cut out what you need. You do not need to spend anything on this as you can use whatever you have around the house and hands-on math should not take you much time to prepare.

Now here are the rules to this game:

• Each tile has an opposite. For this post, white is positive and yellow is negative. It doesn't matter what colors you use, just so it is clear which is which. 
• A zero-pair is formed by pairing one tile with its opposite. 
• You can remove or add zero-pairs without changing the value of the polynomial. 
• Like terms are represented by tiles that are the same shape and size.

Demonstrate how to use the tiles to show each monomial or polynomial.

 Start with 3x to the second power (sorry I don't have any superscript).

 Then demonstrate x to the second power - 2d.

Your student should now be able to do 

2x to the second power + x - 2.

Now use algebra tiles to simplify 2x to the second power + x to the second power + 2x.

Now combine like terms. In its simplest form, 2x to the second power = x to the second power= 2x = 3x to the second power + 2x.

Now use algebra tiles to simplify 3x + 2 - 5x +1. Rearrange the tiles so that like terms are next to each other.

Form zero-pairs, and then remove all zero-pairs.

In its simplest form, 3x + 2 - 5x +1 = 2x + 3.

Math Journal Activities

Now your student should be able to model and simplify any monomial or polynomial that you give him. He can even make up his own problems to solve in his math journal. For some of them, have him sketch a drawing to show how he got his answer. He could also include in his journal a sentence or two to explain how subtracting polynomials is related to adding polynomials.

Hands-on Algebra: Solving Multi-Step Equations

Just as we used cups and counters to solve one-step equations, they can also be used to solve equations with a variable on each side.

For the demonstration problem, 2x + 2 = -4, place 2 cups and two positive counters on one side of the mat. Place 4 negative counters on the other side of the mat. Notice it is not possible to remove the same kind of counters from each side. 

Add 2 negative counters to each side.

Group the counters to form zero-pairs and remove all zero-pairs. Separate the remaining counters into 2 equal groups to correspond to the 2 cups.

Each cup is matched with 3 negative counters. Therefore, x = -3.

Next, demonstrate solving w - 3 = 2w -1 with a cup and counters. Place 1 cup and 3 negative counters on one side of the mat. Place 2 cups and 1 negative counter on the other side of the mat. 

Remove 1 negative counter from each side of the mat.

Just as you can remove the same kind of counter from each side of the mat, you can remove cups from each side of the mat. In this case, you can remove 1 cup from each side.

The cup on the right, or the unknown, is matched with 2 negative counters. Therefore, the answer to the equation is w= -2.

Hands-on Algebra: Solving One-step Equations

Just as we have used cups and beans or chips to learn earlier math concepts, we can use these materials to teach students how to solve one-step algebra equations.

For the purposes of this lesson, a cup represents the variable, white counters represents positive integers and black represents negative integers. After representing the problem with the cup and counters, the goal is to get the cup by itself on one side of the mat by using the following rules:

• A zero-pair is formed by pairing one positive identical counter with one negative counter.
• You can remove or add the same number of identical counters to each side of the equation mat.
• You can remove or add zero-pairs to either side of the equation mat without changing the equation.

For our first teaching problem, we will use the equation x + (-3) = -5.  Place 1 cup and 3 negative counters on one side of the mat. Place 5 negative counters on the other side of the mat. 

Remove 3 negative counters from each side to get the cup by itself.

The cup on the left side is matched with 2 negative counters. Therefore, x = -2.

Now, let us solve the problem, 2p = -6.

Place 2 cups on one side of the mat. Place 6 negative counters on the other side of the mat.

Separate the counters into 2 equal groups to correspond to the 2 cups.

Each cup on the left is matched with 3 negative counters. Therefore, p = -3.

Lastly, let's solve the equation r - 2 = 3.

Let's change the equation to r + (-2) = 3. Place 1 cup and 2 negative counters on one side. Place 3 positive counters on the other side.

Notice that it is not possible to remove the same kind of counters from each side. Add 2 positive counters to each side.

Group the counters to form zero-pairs. Then, remove all zero-pairs.

The cup on the left is matched with 5 positive counters. Therefore, r = 5.

Students can now use what they have learned to solve equations you give them or they can write their own. They can justify their answer with a sketch in their math journals. As a quiz, they can write a paragraph explaining why zero-pairs can be used to solve an equation such as m + 5 = -8.

Hands-on Algebra: Adding and Subtracting Integers

Just as we have used counters to help our students learn addition and subtraction, we can use counters to help them understand addition and subtraction of integers. You will need counters of two different colors, one for positive integers and one for negative integers. We chose green blocks for our positive integers counters and black blocks for our negative integers counters.

The rules for this "game" are as follows:

A zero-pair is formed by pairing one positive counter with one negative counter.

Students can remove or add zero-pair to a set because removing or adding zero does not change the value of the set.

Using these rules, show your student how to use counters to find the sum -3 + (-2).

Place 3 negative counters and 2 negative counters on the mat to symbolize the equation.

Since there are 5 negative counters on the mat, the sum is -5. Therefore, -3 + (-2) = -5. That problem is pretty easy for students to see.

Now, use counters to find the sum -2 + 3.

Place 2 negative counters and 3 positive counters on the mat. Remind your students that it is possible to remove 2 zero-pairs.

Since 1 positive counter remains, the sum is 1. Therefore, -2 + 3 = 1.

Use the counters to find the difference between -4 - (-1).

Place 4 negative counters on the mat. Remove 1 negative counter.

Since 3 negative counters remain, the difference is -3. Therefore, -4 - (-1) = -3.

Use the counters to find the difference between 3 - (-2).

Place 3 positive counters on the mat. There are no negative counters, so you can't remove 2 negatives. 

Add 2 zero-pairs to the mat. Remember, adding zero-pairs does not change the value of the set. Now you can remove 2 negative counters.

Since 5 positive counters remain, the difference is 5. Therefore, 3 - (-2) = 5.

At this point, you can give your student a variety of simple problems that involve adding and subtracting integers, or he can make up some problems of his own. He can solve them using blocks, he can illustrate them in his journal or he can write about how he solved the problems in his math journal.

Hands-on Algebra: The Distributive Property

We have used rectangle tiles to model multiplication. They can also be used to show the Distributive Property of Algebra. 

Use a tile that is about 1 square unit. I am using about a 1-inch square of cardstock. 

Next make an "x" tile by making a unit that is 1 unit wide and as long as you wish. For our purposes, we made it about three or four times as long, but remember that it is "x" units long.

We begin by using the tiles to find the product of 2 (x + 2). The rectangle has a width of 2 units and a length of x + 2 units. We can use our area tiles to mark off the dimensions on a mat, or in this case, a dry-erase board, that will show us the product. Using the marks as a guide, we make the rectangle with the algebra tiles.

The rectangle has 2 x-tiles and 4 1-tiles. The area of the rectangle is x + 1 + 1 + x + 1 + 1 or 2x + 4. Thus, 2(x + 2) = 2x + 4

Now you just need to give your student some practice problems, (or he can even make up some of his own, if he's like.) He can use the tiles and a dry erase board and write the answers in his math journal or he could also solve problems in this way by sketching similar drawings in their math journals.

As a quiz, you could have your student write a paragraph explaining how to find the proof of such problems.

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.