, 1.1 The Counting Numbers 1-1
Chapter 1
Numbers and the Base-Ten System
1.1 The Counting Numbers
1. Answers will vary. For example, when connecting the counting numbers as a list view of
numbers with the number of objects in a set view of numbers, a child must learn to
associate each number in the list in a one to one correspondence with each object in the
set, starting with one. Also, the child must be able to learn that the last number from the
list, used to connect with the last object in the set, is the number of objects in the set.
2. Yes, there is a better way to respond. For instance, you could group the beads into sets of
10 beads in each group. Then you would have 3 groups of 10 beads in each group and
there would be 5 left over beads. This grouping would facilitate a discussion about place
value and allow the conversation to focus on 3 tens.
3. a. You could group the beads into sets of 10 beads in each group. Then you would
have 4 groups of 10 beads in each group and there would be 7 left over beads.
Using the place value system of representing numbers, 4 tens and 7 ones is 47.
Figure 1.1 shows a simple math drawing that could be drawn.
Figure 1.1: Representation of 47
b. You could bag the toothpicks into sets of 10 toothpicks in each bag. Then when
you get 10 bags of 10 toothpicks in each, you could bundle, with a rubber band,
10 bags of 10 toothpicks to make sets of 100 toothpicks in each bundle. Then you
would have 3 bundles of 100 toothpicks in each bundle (or 3 hundreds) and you
would have 2 bags of 10 toothpicks in each bag (or 2 tens) and there would be 8
left over toothpicks. Using the place value system of representing numbers, 3
hundreds, 2 tens, and 8 ones is 328. Figure 1.2 shows a simple math drawing that
could be drawn.
,1-2 Chapter 1: Numbers and the Base-Ten System
Figure 1.2: Representation of 328
c. You could bag the toothpicks into sets of 10 toothpicks in each bag. Then when
you get 10 bags of 10 toothpicks in each, you could bundle, with a rubber band,
10 bags of 10 toothpicks to make sets of 100 toothpicks in each bundle. Then
when you have 10 bundles of 100 toothpicks, you could get a giant gallon sized
plastic bag and put them into it and group these 10 sets of 100 toothpicks into 1
set of 1000 toothpicks. Using the place value system of representing numbers, 1
thousand is represented as 1000. Figure 1.3 shows a simple math drawing that
could be drawn.
Figure 1.3: Representation of 1000
, 1.1 The Counting Numbers 1-3
4.
a. Let’s say you have a relatively unorganized collection of 62 toothpicks as shown
below in Figure 1.4
Figure 1.4: Representation of 62 toothpicks
You cou ld bag the toothpicks into sets of 10 toothpicks in each b ag
See Figure 1.5.
Figure 1.5: Math drawing of 10 ones regrouped as 1 ten.
You could continue doing that with all 62 toothpicks, that is, you can continue
collecting sets of ten toothpicks and regrouping them into tens un til you can no
longer do that. When you are done, you will be able to count the n umber of tens.
If you look at the number 62, the digit in the tens place is a 6 and that corresponds
with thenumber of tens that you find. Similarly, the digit is ones p lace is 2 and
we have 2 single toothpicks not in bundles of ten. See Figure 1.6.
Figure 1.6: 62 represented in base-ten bundles.
b. The 6 in the number 62 could represent 6 tens, as shown in Figure 1.6. If we took the
bundles apart, we would have 60 ones so 6 can stand for that as well. In summary the 6
stands for 6 tens or 60 ones.
5.
a.
You could bag the toothpicks into sets of 10 toothpicks in each bag. Then when
you get 10 bags of 10 toothpicks in each, you could bundle, with a rubber band,
10 bags of 10 toothpicks to make sets of 100 toothpicks in each bundle. Then you
would have 3 bundles of 100 toothpicks in each bundle (or 3 hundreds) and you
would have 2 bags of 10 toothpicks in each bag (or 2 tens) and there would be 8
left over toothpicks. Using the place value system of representing numbers, 3
Chapter 1
Numbers and the Base-Ten System
1.1 The Counting Numbers
1. Answers will vary. For example, when connecting the counting numbers as a list view of
numbers with the number of objects in a set view of numbers, a child must learn to
associate each number in the list in a one to one correspondence with each object in the
set, starting with one. Also, the child must be able to learn that the last number from the
list, used to connect with the last object in the set, is the number of objects in the set.
2. Yes, there is a better way to respond. For instance, you could group the beads into sets of
10 beads in each group. Then you would have 3 groups of 10 beads in each group and
there would be 5 left over beads. This grouping would facilitate a discussion about place
value and allow the conversation to focus on 3 tens.
3. a. You could group the beads into sets of 10 beads in each group. Then you would
have 4 groups of 10 beads in each group and there would be 7 left over beads.
Using the place value system of representing numbers, 4 tens and 7 ones is 47.
Figure 1.1 shows a simple math drawing that could be drawn.
Figure 1.1: Representation of 47
b. You could bag the toothpicks into sets of 10 toothpicks in each bag. Then when
you get 10 bags of 10 toothpicks in each, you could bundle, with a rubber band,
10 bags of 10 toothpicks to make sets of 100 toothpicks in each bundle. Then you
would have 3 bundles of 100 toothpicks in each bundle (or 3 hundreds) and you
would have 2 bags of 10 toothpicks in each bag (or 2 tens) and there would be 8
left over toothpicks. Using the place value system of representing numbers, 3
hundreds, 2 tens, and 8 ones is 328. Figure 1.2 shows a simple math drawing that
could be drawn.
,1-2 Chapter 1: Numbers and the Base-Ten System
Figure 1.2: Representation of 328
c. You could bag the toothpicks into sets of 10 toothpicks in each bag. Then when
you get 10 bags of 10 toothpicks in each, you could bundle, with a rubber band,
10 bags of 10 toothpicks to make sets of 100 toothpicks in each bundle. Then
when you have 10 bundles of 100 toothpicks, you could get a giant gallon sized
plastic bag and put them into it and group these 10 sets of 100 toothpicks into 1
set of 1000 toothpicks. Using the place value system of representing numbers, 1
thousand is represented as 1000. Figure 1.3 shows a simple math drawing that
could be drawn.
Figure 1.3: Representation of 1000
, 1.1 The Counting Numbers 1-3
4.
a. Let’s say you have a relatively unorganized collection of 62 toothpicks as shown
below in Figure 1.4
Figure 1.4: Representation of 62 toothpicks
You cou ld bag the toothpicks into sets of 10 toothpicks in each b ag
See Figure 1.5.
Figure 1.5: Math drawing of 10 ones regrouped as 1 ten.
You could continue doing that with all 62 toothpicks, that is, you can continue
collecting sets of ten toothpicks and regrouping them into tens un til you can no
longer do that. When you are done, you will be able to count the n umber of tens.
If you look at the number 62, the digit in the tens place is a 6 and that corresponds
with thenumber of tens that you find. Similarly, the digit is ones p lace is 2 and
we have 2 single toothpicks not in bundles of ten. See Figure 1.6.
Figure 1.6: 62 represented in base-ten bundles.
b. The 6 in the number 62 could represent 6 tens, as shown in Figure 1.6. If we took the
bundles apart, we would have 60 ones so 6 can stand for that as well. In summary the 6
stands for 6 tens or 60 ones.
5.
a.
You could bag the toothpicks into sets of 10 toothpicks in each bag. Then when
you get 10 bags of 10 toothpicks in each, you could bundle, with a rubber band,
10 bags of 10 toothpicks to make sets of 100 toothpicks in each bundle. Then you
would have 3 bundles of 100 toothpicks in each bundle (or 3 hundreds) and you
would have 2 bags of 10 toothpicks in each bag (or 2 tens) and there would be 8
left over toothpicks. Using the place value system of representing numbers, 3
