,
,
,
, 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 could bag the toothpicks into sets of 10 toothpicks in each bag
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 until you can no
longer do that. When you are done, you will be able to count the number of tens.
If you look at the number 62, the digit in the tens place is a 6 and that corresponds
with the number of tens that you find. Similarly, the digit is ones place 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
Copyright © 2022 Pearson Education, Inc.
,1-4 Chapter 1: Numbers and the Base-Ten System
hundreds, 5 tens, and 8 ones is 328. Figure 1.7 shows a simple math drawing that
could be drawn.
Figure 1.7: Representation of 358
b. The 5 in the number 358 could represent 5 tens, as shown in Figure 1.7. If we took the
bundles apart, we would have 50 ones so 5 can stand for that as well. In summary the 5
stands for 5 tens or 50 ones.
c. The 3 in the number 358 could represent 3 hundreds, as shown in Figure 1.7. If we
took the bundles apart, we would have 30 tens so 3 can stand for that as well. If we took
apart the ten bundles, we would have 300 single toothpicks, so 3 could stand for that as
well. In summary, the 3 stands for 3 hundreds, 30 tens or 300 ones.
6.
For instance, if you we are using Popsicle sticks, you could get a large collection (say
three-hundred fifty-six sticks) in an unorganized pile. Then you could ask the students
that are learning about place value to start counting them. After a bit, someone (you or
some of the students) will likely start to group the Popsicle sticks into piles of equal size.
You could then count the Popsicle sticks faster by bundling groups of 10 together.
Perhaps you might bundle these groups together physically by tying a twist tie around
each set of 10 Popsicle sticks.
After a while of doing this, you will have lots of bundled sets of 10 Popsicle sticks. At
this stage, you could take 10 sets of twist-tied sets of 10 Popsicle sticks and put them in
to a gallon sized plastic bag to make groups of 100 Popsicle sticks (consisting of 10 sets
of 10 bundled Popsicle sticks). As you continue making twist-tied bundles of 10 and
baggies of 100, you eventually will use up all of the Popsicle sticks. When this is done,
you would end up with 3 baggies (or 3 hundreds) and 5 twist-tied bundles (or 5 tens) and
6 left over Popsicle sticks. Now you could point out that your baggies, bundles, and
individual Popsicle sticks correspond directly with the base-ten representation of three
hundred fifty-six Popsicle sticks (or 356). Since the digit in each place value is
representing a count of units that consist of 10 bundles of units that are represented in the
digit to the right, we see that each place value represents a number that is 10 times greater
than the place value to its immediate right. For example, when we count the farthest left
place value in this number (365), we count groups of hundred (3 baggies). Since each
baggie is made up of 10 bundles, we note that the place value immediately to right of the
hundreds place is the place value where we are counting the bundles (or tens).
7. Young children must learn several key ideas about place value and overcome some
linguistic hurdles to learn how to count in the base-ten structure. They must understand
Copyright © 2022 Pearson Education, Inc.
, 1.1 The Counting Numbers 1-5
the key role that 10 plays in our base-ten structure or in representing numbers. They must
understand how the location affects the value of the number or the unit of the digit in that
particular location. They must overcome linguistic difficulties inherent to how we say
numbers in English, especially the anomalies like eleven or the inconsistent order of
twenty-two (2 tens and two ones) and nineteen (one nine and one ten).
8. The number 1001 looks like 100 and 1 put together. Calling it “one hundred one” makes
sense w/o understanding the structure of our number system. See Figure 1.8. Each small
block represents 1.
Figure 1.8: Base-Ten representation of 1001
9. If we count what we’ve got in the math drawing, we see 17 individual toothpicks and 15
bags of 10 toothpicks in each bag. Naively, we might write this as 1517 toothpicks, which
would be misleading since as written it would represent one-thousand five-hundred
seventeen toothpicks.
Since our place value system can only represent up to 9 of any particular place value, we
have to regroup when we have more than 9 of a particular place value (or base-ten unit).
In terms of toothpicks, this means that since 17 is greater than 9, we have enough
individual toothpicks to regroup into one more baggie of 10. This gives us 7 left over
toothpicks but now 16 bags. Similarly, we also have enough bags to regroup (or bundle)
them with a rubber band into one group of 100 toothpicks. This gives us 1 bundle of 100
toothpicks (or 1 hundred), 6 bags of ten toothpicks (or 6 tens) and 7 individual
toothpicks. See Figure 1.9 for what Figure 1.15 (in the regular text) would look like once
you’ve regrouped the numbers in a way that corresponds to the base-ten structure.
Figure 1.9: Representation of 167
Copyright © 2022 Pearson Education, Inc.
,1-6 Chapter 1: Numbers and the Base-Ten System
10. Answers will vary. In the base-ten structure, the digits represent different values of
objects. The place value is integrally related to the value that any particular digit
represents. The number ten plays a vital role in the system and is the basis of the value of
each place. The base-ten structure is much easier to represent large numbers than more
primitive ways of representing numbers. However, the base-ten structure is not as
intuitive and is harder to learn than more primitive systems, such as a simple tally mark
system.
11.
a. You cannot tell where to plot 5 on the given number line. Without another point
for reference, you cannot determine how much space on the given number line
that a unit represents or that some other quantity of units represent.
b. You cannot plot 100 on the given number line. The reasoning is the same as the
answer in part a above.
c. You cannot plot N+1 on the given number line. Without knowing what N
represents, you don’t know how many partitions to make between 0 and N. Those
partition sizes would be each 1 unit, so you cannot determine where one more unit
past N would be.
12.
a. See Figure 1.10. In the first number line I first used the larger tick marks to
represent 400 each. Then I counted up to 800 and then 1200 using these sized tick
marks. Realizing that 900 wasn’t going to fall perfectly on a 400 tick mark, I
partitioned the 400 tick marks spaces into 4 smaller spaces and then each shorter
tick mark represented 100. I then counted one more shorter tick mark past 800 to
get to 900.
b. See Figure 1.10. For the second number line, we partitioned the space between 0
and 300 into 3 equal spaces so between taller tick marks represents 100. Then we
partitioned each of these spaces representing 100 into 2 small spaces representing
50. I counted up to 200 with large tick marks and then over one more smaller
space to 250.
c. See Figure 1.10. For the last number line, I let the spaces between taller tick
marks represent 2000. I went up to 6,000. Next, since it takes ten 200s to make
2,000 and the gap between 6,000 and 8,000 is 2,000, then I made 10 spaces
between 6,000 and 8,000 and plotted 6,200 one of these spaces above 6,000.
Copyright © 2022 Pearson Education, Inc.
, 1.1 The Counting Numbers 1-7
Figure 1.10: Number lines
13.
a. The next two units in base-two after 16 are 32 and 64.
b. See Table 1.1
11 12
13 14
15 16
17 18
19 20
Table 1.1: 11 through 20 in base-two representation
c. 11 = 10112
12 = 11002
13 = 11012
14 = 11102
15 = 11112
16 = 100002
17 = 100012
18 = 100102
19 = 100112
20 = 101002
d. See Figure 1.11. 35 = 1000112.
Figure 1.11: 35 written in base-two.
Copyright © 2022 Pearson Education, Inc.
, 1-8 Chapter 1: Numbers and the Base-Ten System
e. See Figure 1.12. 45 = 1011012.
Figure 1.12: 45 written in base-two.
14.
a. The next two units in base-three after 3 are 9 and 27.
b. See Table 1.2.
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
22 23 24
25 26 27
28
29
30
Table 1.2: Numbers 1 to 30 in base-three math drawings
Copyright © 2022 Pearson Education, Inc.
,
,
, 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 could bag the toothpicks into sets of 10 toothpicks in each bag
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 until you can no
longer do that. When you are done, you will be able to count the number of tens.
If you look at the number 62, the digit in the tens place is a 6 and that corresponds
with the number of tens that you find. Similarly, the digit is ones place 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
Copyright © 2022 Pearson Education, Inc.
,1-4 Chapter 1: Numbers and the Base-Ten System
hundreds, 5 tens, and 8 ones is 328. Figure 1.7 shows a simple math drawing that
could be drawn.
Figure 1.7: Representation of 358
b. The 5 in the number 358 could represent 5 tens, as shown in Figure 1.7. If we took the
bundles apart, we would have 50 ones so 5 can stand for that as well. In summary the 5
stands for 5 tens or 50 ones.
c. The 3 in the number 358 could represent 3 hundreds, as shown in Figure 1.7. If we
took the bundles apart, we would have 30 tens so 3 can stand for that as well. If we took
apart the ten bundles, we would have 300 single toothpicks, so 3 could stand for that as
well. In summary, the 3 stands for 3 hundreds, 30 tens or 300 ones.
6.
For instance, if you we are using Popsicle sticks, you could get a large collection (say
three-hundred fifty-six sticks) in an unorganized pile. Then you could ask the students
that are learning about place value to start counting them. After a bit, someone (you or
some of the students) will likely start to group the Popsicle sticks into piles of equal size.
You could then count the Popsicle sticks faster by bundling groups of 10 together.
Perhaps you might bundle these groups together physically by tying a twist tie around
each set of 10 Popsicle sticks.
After a while of doing this, you will have lots of bundled sets of 10 Popsicle sticks. At
this stage, you could take 10 sets of twist-tied sets of 10 Popsicle sticks and put them in
to a gallon sized plastic bag to make groups of 100 Popsicle sticks (consisting of 10 sets
of 10 bundled Popsicle sticks). As you continue making twist-tied bundles of 10 and
baggies of 100, you eventually will use up all of the Popsicle sticks. When this is done,
you would end up with 3 baggies (or 3 hundreds) and 5 twist-tied bundles (or 5 tens) and
6 left over Popsicle sticks. Now you could point out that your baggies, bundles, and
individual Popsicle sticks correspond directly with the base-ten representation of three
hundred fifty-six Popsicle sticks (or 356). Since the digit in each place value is
representing a count of units that consist of 10 bundles of units that are represented in the
digit to the right, we see that each place value represents a number that is 10 times greater
than the place value to its immediate right. For example, when we count the farthest left
place value in this number (365), we count groups of hundred (3 baggies). Since each
baggie is made up of 10 bundles, we note that the place value immediately to right of the
hundreds place is the place value where we are counting the bundles (or tens).
7. Young children must learn several key ideas about place value and overcome some
linguistic hurdles to learn how to count in the base-ten structure. They must understand
Copyright © 2022 Pearson Education, Inc.
, 1.1 The Counting Numbers 1-5
the key role that 10 plays in our base-ten structure or in representing numbers. They must
understand how the location affects the value of the number or the unit of the digit in that
particular location. They must overcome linguistic difficulties inherent to how we say
numbers in English, especially the anomalies like eleven or the inconsistent order of
twenty-two (2 tens and two ones) and nineteen (one nine and one ten).
8. The number 1001 looks like 100 and 1 put together. Calling it “one hundred one” makes
sense w/o understanding the structure of our number system. See Figure 1.8. Each small
block represents 1.
Figure 1.8: Base-Ten representation of 1001
9. If we count what we’ve got in the math drawing, we see 17 individual toothpicks and 15
bags of 10 toothpicks in each bag. Naively, we might write this as 1517 toothpicks, which
would be misleading since as written it would represent one-thousand five-hundred
seventeen toothpicks.
Since our place value system can only represent up to 9 of any particular place value, we
have to regroup when we have more than 9 of a particular place value (or base-ten unit).
In terms of toothpicks, this means that since 17 is greater than 9, we have enough
individual toothpicks to regroup into one more baggie of 10. This gives us 7 left over
toothpicks but now 16 bags. Similarly, we also have enough bags to regroup (or bundle)
them with a rubber band into one group of 100 toothpicks. This gives us 1 bundle of 100
toothpicks (or 1 hundred), 6 bags of ten toothpicks (or 6 tens) and 7 individual
toothpicks. See Figure 1.9 for what Figure 1.15 (in the regular text) would look like once
you’ve regrouped the numbers in a way that corresponds to the base-ten structure.
Figure 1.9: Representation of 167
Copyright © 2022 Pearson Education, Inc.
,1-6 Chapter 1: Numbers and the Base-Ten System
10. Answers will vary. In the base-ten structure, the digits represent different values of
objects. The place value is integrally related to the value that any particular digit
represents. The number ten plays a vital role in the system and is the basis of the value of
each place. The base-ten structure is much easier to represent large numbers than more
primitive ways of representing numbers. However, the base-ten structure is not as
intuitive and is harder to learn than more primitive systems, such as a simple tally mark
system.
11.
a. You cannot tell where to plot 5 on the given number line. Without another point
for reference, you cannot determine how much space on the given number line
that a unit represents or that some other quantity of units represent.
b. You cannot plot 100 on the given number line. The reasoning is the same as the
answer in part a above.
c. You cannot plot N+1 on the given number line. Without knowing what N
represents, you don’t know how many partitions to make between 0 and N. Those
partition sizes would be each 1 unit, so you cannot determine where one more unit
past N would be.
12.
a. See Figure 1.10. In the first number line I first used the larger tick marks to
represent 400 each. Then I counted up to 800 and then 1200 using these sized tick
marks. Realizing that 900 wasn’t going to fall perfectly on a 400 tick mark, I
partitioned the 400 tick marks spaces into 4 smaller spaces and then each shorter
tick mark represented 100. I then counted one more shorter tick mark past 800 to
get to 900.
b. See Figure 1.10. For the second number line, we partitioned the space between 0
and 300 into 3 equal spaces so between taller tick marks represents 100. Then we
partitioned each of these spaces representing 100 into 2 small spaces representing
50. I counted up to 200 with large tick marks and then over one more smaller
space to 250.
c. See Figure 1.10. For the last number line, I let the spaces between taller tick
marks represent 2000. I went up to 6,000. Next, since it takes ten 200s to make
2,000 and the gap between 6,000 and 8,000 is 2,000, then I made 10 spaces
between 6,000 and 8,000 and plotted 6,200 one of these spaces above 6,000.
Copyright © 2022 Pearson Education, Inc.
, 1.1 The Counting Numbers 1-7
Figure 1.10: Number lines
13.
a. The next two units in base-two after 16 are 32 and 64.
b. See Table 1.1
11 12
13 14
15 16
17 18
19 20
Table 1.1: 11 through 20 in base-two representation
c. 11 = 10112
12 = 11002
13 = 11012
14 = 11102
15 = 11112
16 = 100002
17 = 100012
18 = 100102
19 = 100112
20 = 101002
d. See Figure 1.11. 35 = 1000112.
Figure 1.11: 35 written in base-two.
Copyright © 2022 Pearson Education, Inc.
, 1-8 Chapter 1: Numbers and the Base-Ten System
e. See Figure 1.12. 45 = 1011012.
Figure 1.12: 45 written in base-two.
14.
a. The next two units in base-three after 3 are 9 and 27.
b. See Table 1.2.
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
22 23 24
25 26 27
28
29
30
Table 1.2: Numbers 1 to 30 in base-three math drawings
Copyright © 2022 Pearson Education, Inc.
