Math-Chapter 11 (Place Value)
facile concept of ten
Children are able to easily work with units of ten without the use of physical or mental base-ten models
intermediate concept of 10
Children see ten as a unit, as one thing that consists of ten ones, but they must rely on physical or mental models to help them work with units of ten.
Initial concept of 10
Children understand ten not as a unit but only as ten ones. When solving addition or subtraction problems involving tens, they count only by ones
Contrast this with a child who understands these same quantities in terms of base-ten groupings.
Children who have developed the understanding of base ten concepts though, would see 85 as being 8 groups of tens, and 5 ones. This means they know that the 8 represents the number/groups of tens they have, as it's in the ten-place value, and the 5 represents the number of ones they have, as it's in the ones place value. Finally, if given 85 quantities of something, children would be able to count by groups of tens and then by the ones remaining left over
How would a child who has not yet developed base-ten concepts understand quantities as large as 85?
For a child who hasn't developed base ten concepts, they would see 85 as being 85 single ones. They would not be able to separate the large quantity into place value groups. Some very young children way even see both the 8 and 5 in 85 as representing both single place values. For example, if a question involved beads, they may think that the 8 represents 8 beads and the 5 represents 5 beads. Some children may in this stage may see 85 as just being a number, not realizing 8 and 5 have meaning. A child may also identify the five as five ones, but see the 3 as representing 30 ones, not 3 groups of ten. A child may also know 8 is in the tens position and the 3 is in the ones position but doesn't understand what the positions actually represent. Along with that, if given 85 physical quantities of something, the child would most likely have to physically count these items by one
Three level of understanding the concept of "ten"
Initial concept of 10 intermediate concept of 10 facile concept of ten
What are 3 activities that can be used to help children develop an understanding of large numbers (1000; 10,000; 100,000 and beyond)?
One activity that the book suggested using to help children develop an understanding of large numbers is called Too Many Tens. In this activity, the teacher should show children a quantity of 1000 items, such as a bag of beads. Children can first make estimates of how many beads they think are in the bag and discuss these estimations with the class. Then the teacher can distribute the beads to groups of children who would then put these beads into cups containing ten. Once all groups have done this, collect all the leftovers from all the groups and have the children put them into groups of ten as well. Through questioning that can include, "How can we use these groups of ten to tell how many beads we have," "Can we make new groups from the groups of ten," and "what is 10 groups of ten called." Children can then make groups of hundreds using the already made groups of ten to record the total amount of beads. Then ask the children "How can we use these groups of hundreds to tell how many beads we have," "Can we make new groups from the groups of hundreds," and "what is 10 groups of hundreds called." Children can then record the number on the board. This activity will help reinforce the idea that 1000 is a group of 10 hundreds and 100 is a group of 10 tens Another activity that can be used starts out by writing the standard number name on the board. For example, I could write 1252, which would be the standard name of the number. I could then ask the children to write the base name. The base name would be for this example, 1 thousand, 2 hundreds, 5 tens, and 2 ones. Children can then use a virtual manipulative to represent this number. For example, a student could represent it with 1 thousand piece, 2 hundred pieces, 5 ten pieces, and 2 one pieces. Children can do this with a variety of different numbers, with the virtual manipulative making it possible to represent with higher numbers. This is helping children connect the idea of place values to large numbers, as they are practicing identifying numbers by their place values. The virtual manipulative also helps children actually see what 1252 looks like and how it's made up of 1252 single pieces grouped in different ways. Another activity that can be used starts out by showing children a large number, such as 1,750. This number could also be represented using a virtual representation. I could then ask the children to find at least four ways to represent the number using place value names. They can write this down and manipulate the virtual representation to represent this grouping. For example, 1750 can be written as 1 thousand, 7 hundred, 5 tens, and 0 ones, or 1 thousand, 6 hundreds, 15 tens, and 0 ones. This activity can be done with lager numbers, but the virtual representation may be less applicable the larger the number gets. This is helping children see the relationship between place values and how large numbers can be represented in different ways depending on how they are grouped in terms of thousands, hundreds, tens, and ones.
patterns on a hundreds chart
The numbers in a column all end with the same number, which is the same as the number at the top of the chart. In a row, the first number (tens digit) stays the same and the "second" number (ones digit) counts 1, 2, 3, . . . 9, 0) changes as you move across. In a column, the first number (tens digit) "counts" or goes up by ones as you move down. You can count by tens going down the far right-hand column. Starting at 11 and moving down on the diagonal, you can find numbers with the same digit in the tens and ones (e.g., 11, 22, 33, 44, and so on)
compatible numbers
numbers easily combined to make benchmark numbers Ex: 5, 25, 50, 75
Non proportional models
the ten is not physically ten times larger than the one are not used for introducing place-value concepts. They are used once children have a conceptual understanding of the numeration system and need additional reinforcement.
Four different ways that 37 can be thought about in terms of tens and ones:
1 group of ten and 27 ones 2 groups of ten and 17 ones 3 groups of ten and 7 ones
grouping tens to make 100
100 is a group of 10 tens and 100 ones
Three stages of reasoning related to base ten concept
unitary (counts by ones) base ten (counts by groups of tens and ones) equivalent (can count by groups o tens and ones in a non standard base-ten format)
Real world
as children study place values encourage them to notice the world around them bring real numbers into the classroom
grouping activities
assist children in developing place value and base ten concepts helps children learn to group things by tens and therefore count by tens children can group items and write it's base ten name down Ex: groups straws by ten, write 3 tens and 2 ones, 32
Place value mats
can help promote the concept of groups of ten hundreds spot, tens spot and ones spot children can put their models in the corresponding spots helps children learn to write the number names (have them use place value cards to build the written number as they use the models)
pregrouped models
cannot be taken apart or put together. children have to "exchange" ten ones for ten or 10 tens for a hundred. These groups help model larger numbers It's important to make sure children know that a 10 piece, for example, is still the same as 10 ones even though it can't be taken apart children may just grab 4 tens and 2 ones but don't realize that it actually represents 42 ones
Virtual Base ten blocks
children are able to compose and decompose numbers more quickly with the virtual blocks than with the physical blocks and so were able to generate many more different representations. the virtual site included a place value mat that displayed the number represented by the base-ten blocks. As the base-ten blocks changes, the number remains the same, reinforcing the idea that the representations were indeed equivalent
Equivalent Representations
children are able to represent the same number in different ways. A child, for example, may represent a number by using the base ten language of ones, tens, and so one in different combinations. In other words, this means that 37 can be represented in various combinations of groups of tens and ones. Activities in class then, can be aimed at helping children switch between different number representations, using their knowledge of place values
groupable models
children can literally build one place value piece from another place value piece. This basically means, that the 10-place value can be made or grouped from using 10 single pieces. If students have a cup of 10 skittles, this is the same as 10 single skittles, as you used the single skittles to make the group of ten, representing the ten-place value.
Oral and written names
children must learn these names by being told not through problem based activities
Number Names
connecting base ten concepts with the oral number name 47 (standard name) goes to 4 tens and 7 ones and vice versa could sow children a model and they can give you the base ten name and standard name can do the same with three digit numbers as well
base ten concept
grouping by ten
Place-value understanding requires an integration of new and sometimes difficult-to- construct concepts of
grouping by tens (the base-ten concept) with procedural knowledge of how groups are recorded in our place-value system and how numbers are written and spoken.
hundreds chart
important tool in the development of place-value concepts there are lots of patterns on a hundreds chart
How to integrate base ten grouping with words?
instead of just saying 35, which is the standard name, can say.... 3 tens and 5 ones 3 groups of ten and 5 ones 3 tens and 5 singles you are referring to a number with its place value location not just a digit
Having a relational understanding of base ten means?
integrating base ten concepts, oral names, and written names for numbers
addition and subtraction are good ways to
learn place value concepts children can use their knowledge of benchmark numbers and place value to develop flexible computational methods
Written Symbols
make sure children understand what the symbols actually represent
Proportional models
models where the model for ten is physically ten times larger than the model for one can be groupable or pregrouped
Bench mark numbers
multiplies of 10, 100, and sometimes numbers like 25 understanding how numbers are related to these special numbers is an important step in children's development of number sense and place-value understanding
How to integrate base ten grouping with counting by ones?
once children have counted something by ones help them see making groups of tens and leftovers is the same quantity through questioning
Number sense is linked to an understanding of
place value and base ten number system
Place value can be developed by children.....
putting numbers together and taking them apart as they solve addition and subtraction problems (composing and decomposing)
Physical models
support the development of place value and base ten understanding
common misconceptions about place value
writes numbers in an expanded form that mirrors the place value words reverses digits when writing represents the number with base ten blocks using the face value of the digits (for 13 only uses 1 blocks and 3 blocks) When given an arrangement of base-ten blocks to count, the child ignores the 10-1 relationship between the pieces and uses the same count for different pieces. The child is easily confused about the meaning of the digits in a multi-digit number.
How can you integrate base ten groupings with place value notation?
you can use activities where children have to associate groups of tens and ones with the correct place value Ex: cubes can be grouped by tens and ones and children can put these numbers into a chart labeled tens and another column labeled ones, which is it's notation
It's important to emphasize......
zero in place value
