When my college students (remedial math students) finish the first chapter in
Fractions, Decimals, and Percents, we focus on place value. Over the years, I have come to the realization how vital it is to provide a careful development of the basic grouping and positional ideas involved in place value. An understanding of these ideas is important to the future success of gaining insight into the relative size of large numbers and in computing. A firm grasp of this concept is needed before a student can be introduced to more than one digit addition, subtraction, multiplication, and division problems. It is important to stay with the concept until the students have mastered it. Often when students have difficulty with computation, the source of the problem can be traced back to a poor understanding of place value.
It was not surprising when I discovered that many of my students had never used base ten blocks to visually see the pattern of cube, tower, flat, cube, tower, flat. When I built the thousands tower using ten one hundred cubes, they were amazed at how tall it was. Comparing the tens tower to the thousands tower demonstrated how numbers grew exponentially. Another pattern emerged when we moved to the left; each previous number was being multiplied by 10 to get to the next number. We also discussed how the names of the places were also based on the pattern of: name,
tens,
hundreds, name (thousands),
ten thousands,
hundred thousands, etc.
I asked the question,
"Why is our number system called base ten?" I got the usual response,
"Because we have ten fingers?" Few were aware that our system uses only ten digits (0-9) to make every number in the base ten system.
We proceeded to look at decimals and discovered that as we moved to the right of the decimal point, each number was being divided by 10 to get to the next number. We looked at the ones cube and tried to imagine it being divided into ten pieces, then 100, then 1,000. The class decided we would need a powerful microscope to view the tiny pieces. Again, we saw a pattern in the names of each place:
tenths,
hundredths, thousandths,
ten thousandths,
hundred thousandths, millionths, etc.
I then got out the Decimal Show Me Boards. (See illustration on the left.) These are very simple to make. Take a whole piece of cardstock (8.5" x 11") and cut off .5 inches. Now cut the cardstock into fourths (2.75 inches). Fold each fourth from top to bottom. Measure and mark the cardstock every two inches to create four equal pieces. Label the sections from left to right - tenths, hundredths, thousandths, ten thousandths. You can type up the names of the places which then can be cut out and glued onto the place value board.
Here are some examples of how I use the boards. I might write the decimal number in words. Then the students make the decimal using their show me boards by putting the correct numbers into the right place. Pairs of students may create two different decimals, and then compare them deciding which one is greater. Several students may make unlike decimals, and then order the decimals from least to greatest. What I really like is when I say,
"Show me", I can readily see who is having difficulty which allows me to spend some one-on-one time with that student.
Show Me Boards can also be made for the ones, tens, hundreds and thousands place. Include as many places as you are teaching. I've made them up to the hundred thousands place by using legal sized paper. As you can see in the photo above, my two granddaughters love using them, and it is a good way for them to work on place value.
A good way to practice any math skill is by playing a game. Your students might enjoy the No Prep place value game entitled:
Big Number. Seven game boards are included in this eleven page resource packet. The game boards vary in difficulty beginning with only two places, the ones and the tens. Game Board #5 goes to the hundred thousands place and requires the learner to decide where to place six different numbers. All the games have been developed to practice place value using problem solving strategies, reasoning, and intelligent practice.
A kite has just one pair of equal angles. These congruent angles are a light orange on the illustration on the left. A kite also has one line of symmetry which is represented by the dotted line. (A line of symmetry is an imaginary line that divides a shape in half so that both sides are exactly the same. In other words, when you fold it in half, the sides match.) It is like a reflection in a mirror.
The diagonals of the kite are perpendicular because they meet and form four right angles. In other words, one of the diagonals bisects or cuts the other diagonal exactly in half. This is shown on the diagram on the right. The diagonals are green, and one of the right angles is represented by the small square where the diagonals intersect.
