Place Value: Application and Extension (Pre K-2)

Estimating and Checking

Present to your students the task of estimating how many objects are in any empty container you have. You can use any math manipulative for this activity, but I like to use snacks, such as these gummy frogs or popcorn or peanuts in the shell, because it interests children.

Then ask one student to take the counters out of the container and begin counting them out on a Place Value Counting Board. This is a pleasureable way to learn estimating skills if you do this regularly with different counters and different containers of all sorts of shapes and sizes.

After you have done this several times, you can increase the amount of counters you are usings as well. Initially their estimates will not be so accurate this time, as the numbers you work with are higher. Give your students chances to change their estimates as they go along because the object is for them to get better at estimating, not winning against each other. Once you have completed your estimating exercise and are ready to hand out the counters for a snack, first have them figure out the quickest and easiest way to divide the crackers equally.

Peas in the Pod

First show your students unshelled peas and ask them how many peas on the average are in each pea pod. Record their guesses. You can make a graph of their predictions.

Now open the pea pods and have them count how many peas are on each side, counting one side at a time and recording the numbers as an addition problem. (Number of peas on left side plus number of peas on the right side equals how many peas altogether.) Put the peas in a cup and place the cup on a graph according to the total number of peas in the pod. Repeat this until all the pods have been opened and graphed. How close was the average to the predictions?

If you have older children, have them figure out the average number of peas in pods by adding together all the answers to the equations and dividing by the number of equations you did. Was the average the same as the average on the graph?

Perimeters

For this activity, you will need records of tile and geoboard designs from exploring numbers at the number stations.

Students place Unifix cubes around the perimeter of the base design, snap the cubes together into sticks of tens, count them and record the total. This is repeated many times for the various designs of six to ten.

Geoboard Designs

The student makes a shape on the geoboard using one large band. At first students can cover the pegs inside the one shape with one color of Unifix cubes and the pegs on the outside with a second color Unifix cubes. They then take the Unifix cubes off and record each number and the total number of pegs on their paper. (Worksheet for this can be found here.) After they have become more accomplished with counting, they can just count the pegs and record their numbers as addition problems of their own making.

The Store

You will need to pick out about ten to fifteen common store-bought items with the prices clearly marked. If they do not have the prices on them, use stickers to price them yourself.

 Also print out some blank addition cards.   Students can pick two items and write down their prices. If they also write the name of the item, it is easier to check their answers later. Students can imagine they have a dollar to spend and them subtract each item from the dollar to determine the change. You can also use real money for this game, but it is not necessary. (More about an example of a store game here.)

I will be adding more activities as we do them:

The Movie Theatre

Determining Prices

Base Ten Unifix Patterns

Counting Jars of Objects

Estimating and Graphing

Unifix Stacks

Recording Number Patterns from Row, Column and Diagonal Patterns with Unifix Cubes

Recording Number Patterns from Surrounding Patterns

Measuring

source: Mathematics Their Way, Mary Baratta-Lorton

Place Value: Introduction: The Counting Game (Pre K-2)

Concept Development

"Zero spocks and one."

This counting game activity gives students a understanding of place value (base 10) by exploring groupings in bases other than ten and look for the patterns. Once they can see and understand the patterns of smaller groupings, it is easier to understand the same patterns in base ten.

I begin by asking for a silly word or fun game. This makes the game more appealing. Quentin suggested spock, but have your student come up with his own silly word. It can be a totally made up word like zurkle or bosco or bloop. I then explained the rules to the game. The first rule is that the blue column is where we are going to put spocks. The green column is where we add one counter at a time. As soon as we have enough for a Spock, we exchange the green counters for one blue spock counter. You determine in advance what base you want to explore of this day and explain to him that once you get to a certain amount, it becomes one of your silly words and goes in the blue column. In this case, I chose for us to explore base 4, so every time we had added one more to three in the green column, we exchange the four greens for 1 spock.. For each round, the student should read the board, saying the number of spocks first and then the number of ones. You can ring a bell to signal the start of a new round or you can just say "new round,' whichever would appeal to your student more.

The picture above is the result of round one, and Quentin read it as "zero spocks and one."

"Zero spocks and two."

This is the result of round two. Continue on in this way until you get to the base you want to explore.

"Zero spocks and three."

"Spock!"

"One spock, and zero."

 Make sure that your student exchanges the four counters for one spock.

Keep playing the game, counting one at a time up to the point that you would need to exchange the counters in the blue column for one in the next (red) column. Now start taking one away for each round. Make sure your student repeats the name for each step.

"Three spocks and two."

Play this game as long as you have his interest. The younger your student, the more days you might have to play it, but play it until your student can easily add and subtract by one and regroup without being instructed.

 Once you reach that point, you are ready to start the game all over again with a new  amount, and a new fun word. You can also change materials because this encourages flexible thinking.

Repeat the game with various groupings and names until they can anticipate the adding and subtracting.

Quentin is practicing counting forwards and backwards in base 6 using cups and beans. 

After some time, perhaps weeks or even months, and your student has played several grouping games and you feel they really understand the pattern, proceed to connecting symbols to the concept.

Connecting Symbolization to the Concept

Using Base 5, we set our counters, with the line down the middle representing the same division as the blue and green divisions above. On this particular day, they were called "Googas.'

 We counted directly on the board with the ones on one side and the googas on the other.
We then wrote numerically on the side.

We also use number-flips, which can be as simple as homemade numeral cards made from index cards cut in half. As they count up and down the base they are practicing, they flip over to make the numeral match the amounts on their counting boards.

Using Symbols to Record the Concept

 Once you feel they can  confidently count up and down a variety of bases, you can begin the recording level of this game. Have them continue to add one, of whatever manipulative you wish to use, at a time to their boards, but instead of flipping to the appropriate numeral, ask them to read their boards and you can add this to a strip of 2-column block paper. Repeat this activity with several other base grouping and he should be able to do the recording page independently before long. When they finish a recording page, look together for patterns. 

To introduce variety in making written records, introduce a square array chart. If you are doing sixes, use a 6 x6 graph grid, if you are doing base four, use a 4 x 4 graph grid and so on.  Have them copy the sequence and then color the patterns they see, such as where do all the same numerals fall in the different place value columns? For example, where do all the 1's fall in the ones column or the bloops (or whatever word you are using) column? 

Pattern Two: Introduction (Pre K-2)

Fruit and Vegetable Patterns

This activity is sometimes easier to do if you can find fruits or vegetables in which the students have not seen before, at least not on the inside. This is so they have to think about what they might look like inside based solely on what they see on the outside. Have them sketch the outside of the fruits so that they would pay close attention to their shape, size, texture and color.

Then they sketched what they guessed the inside would look like. If the outside was tough, would that make the inside soft? Would their be cavities inside? Would the color be the same or different? If the color was the same, would the shade be different? What would the seeds look like? Would they be small or large? Hard or soft? I made sure that they knew that I did not expect them to get the correct answers, but that I just wanted them them wonder, to infer.

Then we cut them open to see how they looked.

And they drew them again, noting all the things that they had wondered about before.

I encourage them to describe the shape, texture and markings of each of them. Note the color. Can you make your sketch have the exact color of the fruit or vegetable? What is the texture like -smooth, bumpy, crackly like paper? Does it pull apart? Does it have spots or other markings? How many sections does it divide into?

Using a magnifying glass can help for closer examination.

Can they predict what each piece will look like when it is cut open? Will the color be the same on the inside as on the outside? Do you expect to find a few seeds or a great many or none at all?

Discuss the lines and the proportion and the different shades of color.
Cut it open and talk about what you discover. Look at the different patterns inside each piece. Do you see spiraling patterns?

Leaf Patterns

Have your student examine a branch of leaves and ask them to describe whatever they notice. Do you see different shades of color? Do you see any places where there are different shades of color? Do you see any places where there are different thicknesses? Do these always occur in the same place? what do you notice about the spaces between the leaves? Are all the leaves the same size? How are they attached to the branch? Are they like your arm and are straight across or do they zig-zag? Do you see any diagonals? Are all the parts the same texture? The leaves on many plants are staggered in a spiral pattern to permit optimum exposure to sunlight. If we apply the Golden Ratio to a circle we can see how it is that this plant exhibits Fibonacci qualities. Cut the leaves off the branch. Can they sort them?

Make an effort to encourage them to notice patterns during nature study. Encourage them to make as accurate a sketch as they are capable, or take photographs to examine later for patterns.

Clothing Patterns

Students examine the design on a piece of fabric and copy this pattern onto paper.

Routes

You will need an enlarged map of your neighborhood for this project. The student picks a destination and plots the different ways to get there. It is fun to check their directions by going on a walk to the destination, using their directions.

Tile Patterns

Your student first makes a design using four blocks, tiles or Unifix cubes. Then ask your student how he could add tiles to keep the design, but adding to it, evenly. Have the student predict and build until five or six steps of the pattern are completed.

Surrounding Patterns

The couple of squares with the x's in them were mistakes.
This is why it  is good to check  the blocks before your students color the blocks in.

Have your student make a design with five or six blocks or cube on a piece of graph paper. Have him remove the blocks, one at a time, and color the design on the graph paper. The student now "surrounds" the design by placing a block in each space that touches only one side of the base design. Once you have checked to see if this rule has been adhered to, the student then colors these blocks, taking off one at a time. Continuing doing this until either the design is played out or the student tires of the activity.

Row, Column and Diagonal Patterns with Unifix Cubes

A lot of mathematic observations can be made by looking at the patterns numbers make. Using graph paper with different size blocks, have your students color according to number patterns. First have them build the patterns using Unifix cubes. For example, the number two would have two colors in an A-B-A-B pattern. Have them then pick one of the colors and trace the patterns they make in the blocks with a black marker.

Notice how James would color each block of the pattern in order.

Sam would get the pattern and color whole sections, one color at at time

Alex would color one color at a time.

Names

Students write their names in square number arrays, beginning each one with the first letter of their name and leaving no spaces between letters or names.

Then they color in the first letter of their names and look for patterns.

Numbers at the Concept Level: Introduction: The Number Stations (Pre K-2)

Tiles or Unifix Cubes

In this game each student uses two colors of bathroom tiles or Unifix Cubes to make different patterns with the number being explored.

Junk Boxes (or even a snack!)

On another day, they explored the number "5," and they wanted to use goldfish, but we only had one color of them, so instead of arranging patterns, their task was to arrange 5 goldfish in various different ways. In this way they can get an idea of what 5 looks like in various arrangements.

Beans and Pasta

Using two-sided beans (beans sprayed painted on one side) or different types of pasta, students shake them in a cup and spill them out on the table. What different patterns can be found? It is interesting to see how this game can continue to keep interest as they make new discoveries with different amounts of materials. We vary the types of materials as well.

Sam and James showed the concept that 7 is an odd number by showing two sets of 3, with the odd man out in the middle.

James showed the concept that we naturally can only mentally conceive of 5 and then after that branch off to adding more to 5. He had a row of 5 with 2 branching off, or 5 +2=7

Quentin is most comfortable with the number 4. Even though often he can successfully count to 10, he still becomes unsure after 4, so he split his 7 in 4+3=7

Geoboard

Here are the results of an exploration of five.



You will need to make up some squares of cardstock (or index cards) that fit between the pegs on your geoboard. Your student then can explore a given quantity and find as many different arrangements. The only rule is that each square must be adjacent to another square (corners are fine.)

I will be adding more activities as I can:

Toothpicks

Tiles

Pattern Blocks

Jewels

Recording Number Stations

Tile Patterns and Geoboard Patterns

To record Tile/Unifix Cube Patterns and Geoboard Patterns, have available pre-cut squares of colored paper the same size as your tiles/Unifix cubes. Students glue these squares onto 6 x 9 inche pieces of white paper in the design of their choice. Another option is to have available graph paper with large squares and have student color in the squares to copy the designs they have made.