Name _____________________________________ Date ____________ Period ___________
Meter Stick Building
Stage 0
1. Object being estimated: _________________ Your fraction estimate: ____
2. Why was it difficult to compare your estimate with your classmates? Why
would it be difficult to find who had the closest estimate?
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
Stage 1
3. With your meter paper marked, estimate the length of the object to the nearest
tenth. Write this as both a fraction and a decimal.
Estimate in fraction form: ______
Estimate in decimal form: ______
4. Color parts of the diagram below to represent your estimate:
5. How many parts did you color? ____ What is the length of each part? ____
6. Write these two numbers next to each other using multiplication: _________
This is a third equivalent way to write your estimate.
7. After measuring, what is the length of the object to the nearest tenth? _____
8. How many tenths was your estimate away from the measured length? (Answer
in a complete sentence.)
____________________________________________________________________________
____________________________________________________________________________
IMP Activity: Meter Stick Building
1
PVD S1
Stage 2
9. How many parts does your meter strip have now? _____
10. What are the two ways to write the name of these little parts? ____
_____
11. Estimate the length of the object to the nearest hundredth. Write this as both a
fraction and a decimal.
Estimate in fraction form: ______
Estimate in decimal form: ______
12. Put a pencil mark on your paper strip at your estimate..
13. Count from the 0 end of the strip. How many complete tenths are there before
you get to your estimate? ____ How many extra hundredths? ____
14. Write the number of tenths ×
1
10
plus the number of extra hundredths ×
1
100
:
_________________________________
This is a third equivalent way to write your new estimate.
15. After measuring, how long is the object to the nearest hundredth? _____
16. How many hundredths off was your estimate? (Answer is a complete
sentence.)
_____________________________________________________________________________
_____________________________________________________________________________
Stage 3
17. Compare your paper strip to a meter stick. What does the meter stick have that
your paper strip does not?
__________________________________________________________________
__________________________________________________________________
18. How many parts are there on the meter stick? ___________
19. What are each of these parts called? ______________
IMP Activity: Meter Stick Building
2
PVD S2
20. Estimate the length of the object to the nearest thousandth. Write this as a
decimal and a fraction. Label each place value in the decimal.
0 . ___ ___ ___
21. Write in words how we would read the fraction above:
__________________________________________________________________
This is the same way that we read the decimal above.
22. What do you notice about which place value we say when we are reading a
decimal?
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
23. After measuring, how long is the
object to the nearest thousandth? Write
the length here and label each of the
place values.
0 . ___ ___ ___
24. How many thousandths off was your estimate? (Answer in a complete
sentence.)
_____________________________________________________________________________
_____________________________________________________________________________
IMP Activity: Meter Stick Building
3
PVD S3
Name _____________________________________ Date ____________ Period ___________
Meter Sticks and Expanded Decimals Using your meter paper.
• Find an object in the room that is longer than 0.1 meters but shorter than 1
meter.
• Use your paper meter strip to measure it. First find how many full tenths fit
within the length. Next find how many extra hundredth there are.
• Select the decimal card that has the number of tenths.
• Select the decimal card that has the number of extra hundredths.
• Layer the tenths card on top of the hundredths card to see the standard
decimal form.
• Record the name of the object, the tenths, the hundredths and the standard
decimal form below.
• Repeat for 3 more objects.
Object 1 Name: ____________________
___________ + ____________ =
tenths
hundredths
______________
standard decimal form
Object 2 Name: ____________________
___________ + ____________ =
tenths
hundredths
______________
standard decimal form
Object 3 Name: ____________________
___________ + ____________ =
tenths
hundredths
______________
standard decimal form
Object 4 Name: ____________________
___________ + ____________ =
tenths
hundredths
______________
standard decimal form
!
IMP Activity: Meter Sticks and Expanded Decimals
1
PVD S4
Using a meter stick.
• Find an object in the room that is longer than 0.1 meters but shorter than 1
meter.
• Use a meter stick to measure it. First find how many full tenths fit within the
length. Next find how many extra hundredths will fit. Finally, read how
many extra thousandths there are.
• Select the decimal cards that match the number of tenths, hundredths and
thousandths.
• Layer the three cards to see the standard decimal form.
• Record the name of the object, the tenths, the hundredths, the thousandths
and the standard decimal form below.
• Repeat for 3 more objects.
Object 5 Name: ____________________
___________ + ___________ + ___________ =
tenths
hundredths
thousandths
______________
standard decimal form
Object 6 Name: ____________________
___________ + ___________ + ___________ =
tenths
hundredths
thousandths
______________
standard decimal form
Object 7 Name: ____________________
___________ + ___________ + ___________ =
tenths
hundredths
thousandths
______________
standard decimal form
Object 8 Name: ____________________
___________ + ___________ + ___________ =
tenths
hundredths
thousandths
!
IMP Activity: Meter Sticks and Expanded Decimals
______________
standard decimal form
2
PVD S5
Estimating to the nearest hundredth.
Object 9 Name: ____________________
___________ + ____________ =
tenths
hundredths
______________
standard decimal form
Object 10 Name: ____________________
___________ + ____________ =
tenths
hundredths
______________
standard decimal form
Estimating to the nearest thousandth.
Object 11 Name: ____________________
___________ + ___________ + ___________ =
tenths
hundredths
thousandths
______________
standard decimal form
Object 12 Name: ____________________
___________ + ___________ + ___________ =
tenths
hundredths
thousandths
______________
standard decimal form
Bonus:
Which object did you estimate the best? Show your estimate and the measured
value below. Use any method you can to find how far apart your estimate was
from the measured value.
!
IMP Activity: Meter Sticks and Expanded Decimals
3
PVD S6
!
IMP Activity: Meter Sticks and Expanded Decimals
4
PVD S7
.0 0 1
.0 0 2
.0 0 3
.0 0 4
DECIMAL CARDS PVD S8
.0 0 5
.0 0 6
.0 0 7
.0 0 8
DECIMAL CARDS PVD S9
.0 0 9
.0 1
.0 2
.0 3
DECIMAL CARDS PVD S10
.0 4
.0 5
.0 6
.0 7
DECIMAL CARDS PVD S11
.0 8
.0 9
.1 .2 .3 .4 DECIMAL CARDS PVD S12
.5 .6 .7 .8 .9 DECIMAL CARDS PVD S13
Name: ________________________________________
Date:________________
Rounding and Estimating Decimals
Opening Scenario:
You are helping your teacher to prepare materials for a science lab that uses many wires cut to
different lengths. You have a new coil of wire that says it has 4 meters on it. The wires that will
need to be cut are listed below.
0.88 m
0.092 m
0.541 m
0.08 m
0.604 m
0.007 m
0.240 m
0.5 m
0.78
Your teacher is headed to the store room and asks, “Will there be enough wire or do I need to
bring another coil?” You don’t have time to grab a pencil or paper or a calculator and need to
decide if you’ll have enough. What do you say?
Do you have enough wire? _______________________________________________
How do you know? Explain your thinking. __________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Estimating/Rounding
The methods the class used to determine if you had enough wire involved estimating and
possibly rounding. When we don’t need an exact answer, rounding can help us estimate
quickly.
Define rounding in your own words: _______________________________________________
____________________________________________________________________________
Define estimating in your own words: ______________________________________________
____________________________________________________________________________
IMP Activity Rounding and Estimating Decimals
1
PVD S14
Rounding - Part 1
Use your meter stick to find and record the halfway point between the numbers
given.
1. What point is halfway between 0.2 and 0.3? ____________
2. What point is halfway between 0.5 and 0.6? ____________
3. What point is halfway between 0 and 0.1? ____________
4. What point is halfway between 0.25 and 0.26? ____________
5. What point is halfway between 0.47 and 0.48? ____________
6. What point is halfway between 0.9 and 1? ____________
7. What point is halfway between 0.79 and 0.8? ____________
8. What point is halfway between 0.4 and 0.41? ____________
Explain as best you can why the halfway point between two numbers (also called
the midpoint) is important when rounding.
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
IMP Activity Rounding and Estimating Decimals
2
PVD S15
Rounding - Part 2
When rounding, if your number is halfway or more to the next number of
the place value to which you are rounding, round UP to the next value. If
the number is less than halfway, go back to the last value.
Use your meter stick to locate each number. Then round it to the required place
value.
1. Round 0.37 to the nearest tenth: ________________
2. Round 0.84 to the nearest tenth: ________________
3. Round 0.98 to the nearest tenth: ________________
4. Round 0.261 to the nearest tenth: ________________
5. Round 0.439 to the nearest tenth: ________________
6. Round 0.275 to the nearest hundredth: ________________
7. Round 0.682 to the nearest hundredth: ________________
8. Round 0.934 to the nearest hundredth: ________________
9. Round 0.297 to the nearest hundredth: ________________
10. Round 0.703 to the nearest hundredth: ________________
IMP Activity Rounding and Estimating Decimals
3
PVD S16
Name _____________________________________ Date ____________ Period ___________
Three Meter Dash Your goal is to be the first to cross the finish line three meters away. In each round you will roll
a die to get the digits of a decimal number. After each roll you must decide where to put the
number from the die. Three rolls will be used and one won’t.
Round 1
Roll a single die. Write the digit in one of the places below. Repeat this three more times to fill
in all of the spaces below.
Who moved farther in Round 1?
0. ___ ___ ___
Unused number ___
You cannot move a digit after you write it.
Move your post-it this distance toward the finish.
0. ___ ___ ___
me
0. ___ ___ ___
my opponent
Use <, > or = to compare.
Round 2
Roll a single die. Write the digit in one of the places below. Repeat this three more times to fill
in all of the spaces below.
Who moved farther in Round 2?
0. ___ ___ ___
Unused number ___
You cannot move a digit after you write it.
Move your post-it this distance toward the finish.
0. ___ ___ ___
me
0. ___ ___ ___
my opponent
Use <, > or = to compare.
Round 3
Roll a single die. Write the digit in one of the places below. Repeat this three more times to fill
in all of the spaces below.
Who moved farther in Round 3?
0. ___ ___ ___
Unused number ___
You cannot move a digit after you write it.
Move your post-it this distance toward the finish.
0. ___ ___ ___
me
0. ___ ___ ___
my opponent
Use <, > or = to compare.
Round 4
Roll a single die. Write the digit in one of the places below. Repeat this three more times to fill
in all of the spaces below.
Who moved farther in Round 4?
0. ___ ___ ___
Unused number ___
You cannot move a digit after you write it.
Move your post-it this distance toward the finish.
!
IMP Activity: Three Meter Dash
0. ___ ___ ___
me
0. ___ ___ ___
my opponent
Use <, > or = to compare.
1
PVD S17
Use as many rounds as is needed until someone crosses the finish line.
Round 5
Roll a single die. Write the digit in one of the places below. Repeat this three more times to fill
in all of the spaces below.
Who moved farther in Round 5?
0. ___ ___ ___
Unused number ___
You cannot move a digit after you write it.
Move your post-it this distance toward the finish.
0. ___ ___ ___
me
0. ___ ___ ___
my opponent
Use <, > or = to compare.
Round 6
Roll a single die. Write the digit in one of the places below. Repeat this three more times to fill
in all of the spaces below.
Who moved farther in Round 6?
0. ___ ___ ___
Unused number ___
You cannot move a digit after you write it.
Move your post-it this distance toward the finish.
0. ___ ___ ___
me
0. ___ ___ ___
my opponent
Use <, > or = to compare.
Round 7
Roll a single die. Write the digit in one of the places below. Repeat this three more times to fill
in all of the spaces below.
Who moved farther in Round 7?
0. ___ ___ ___
Unused number ___
You cannot move a digit after you write it.
Move your post-it this distance toward the finish.
0. ___ ___ ___
me
0. ___ ___ ___
my opponent
Use <, > or = to compare.
Who won the three meter dash?
Winner’s Distances
__________________________
0. ___ ___ ___
0. ___ ___ ___
Check the winner’s total distance by using a
calculator to add all of the distances from each of
the rounds. The total should be greater than 3
meters.
0. ___ ___ ___
0. ___ ___ ___
0. ___ ___ ___
0. ___ ___ ___
0. ___ ___ ___
Total = _________________
!
IMP Activity: Three Meter Dash
2
PVD S18
203
203 203
204
204
blocks
canacan
be
abe
powerful
toolseeing
for seeing
these
representations.
instance,
Base-10
blocks
can
be
powerful
tool for
representations.
For For
instance,
if if if
204Base-10
Base-10
blocks
a powerful
tool
for these
seeing
these
representations.
For
instance,
205
205
is used
to represent
1 whole
(the
whole
or unit),
then
a “stick”
represents
1/10,
and
a a
a “flat”
isName
toisrepresent
1 (the
orwhole
unit),
then
a “stick”
1/10,
and
a and
205a “flat”
aused
“flat”
used
to represent
1 (the
or
unit),
then
arepresents
“stick”
represents
1/10,
_____________________________________
Date
____________
Period
___________
206
206small
small
“cube”
represents
1/100.
Students
can
be challenged
tosense
make
sense
a number
206
“cube”
represents
1/100.
Students
canchallenged
be challenged
to make
sense
aofnumber
small
“cube”
represents
1/100.
Students
can be
to make
of
aofnumber
207
Base 10 Blocks and ecimal P23/100:
lace Value 0.23
as
being
represented
by
both
2/10
+and
3/100
and
23/100:
207
like like
0.23
as being
represented
by
both
2/10
+D3/100
and
like207
0.23
as
being
represented
by
both
2/10
+
3/100
23/100:
208
208 208
You may have worked with Base 10 blocks before. Here are the basic pieces:
If
If
If
represents
1,represents
then represents
represents
andarepresents
represents
represents
1, then
.
represents
then
and and
represents
.
a1,flat
a rod
and
cube
.
“Explain
why
theboth
following
represent
the0.23”
number
“Explain
why
the following
bothboth
represent
the number
0.23”0.23”
“Explain
why
the
following
represent
the number
1. Explain what you know or can see about the relationships between the pieces.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
I that
see
that
thehundredths
20 hundredths
in picture
the
on can
the right
can
be grouped
into
sets
10 hundredths.
I seethe
20
the
onright
the
right
cangrouped
be grouped
2 sets
ofhundredths.
10ofhundredths.
“Well, “Well,
I see“Well,
that
20the
hundredths
in theinpicture
onpicture
the
be
into 2into
sets
of210
means
these
2 groups
represent
2 tenths,
or
. There
3 hundredths
left,
so altogether
ThatThat
means
these
2 unit
groups
represent
2 tenths,
. There
3are
hundredths
left,altogether
so
altogether
are are
That means
these
2 groups
represent
2 decimals,
tenths,
or
. or
There
are
3are
hundredths
so
therethere
arethere
Because
this
is about
we
will
use
the
blocks
in aleft,
special
way.
Throughout
+
.”+
this
the large flat piece will represent one whole.
+ unit,
.”
.”
2. Explain what each of the other pieces represents.
209
209 209
210
_________________
210Students
Students
need
to understand
size
of decimal
numbers
and
relate
to common
_________________
210
to understand
the the
size
of decimal
numbers
relate
them
to common
Students
needneed
to understand
the size
of decimal
numbers
and and
relate
them
tothem
common
211
211benchmarks
benchmarks
such
as
0, 0.5
(0.50
0.500),
and
1. Comparing
tenths
to tenths,
211
as
0.5
(0.50
andand
0.500),
and
1. Comparing
tenths
to _________________
tenths,
benchmarks
such such
as 0,
0.50,
(0.50
and
0.500),
and
1. Comparing
tenths
to
tenths,
212
212hundredths
hundredths
to hundredths,
thousandths
to thousandths
is simplified
if students
212
to hundredths,
andand
thousandths
to thousandths
is simplified
if students
hundredths
to hundredths,
and thousandths
to thousandths
is simplified
if students
useuseuse
213
213understanding
understanding
of fractions
to compare
decimals.
213
theirtheir
understanding
of fractions
to compare
decimals.
their
of fractions
to compare
decimals.
214
214 214
= 1 whole
_________________
_________________
_________________
3. Gather blocks and then draw pictures to represent each of the following numbers.
Example:
Example:
Example:
0.27
b)might
0.30
d)to 0.41
Comparing
0.207
to 0.26,
a student
think,
“Both
numbers
have
2sotenths,
need
to compare
Comparing
to a
0.26,
a student
might
think,
“Both
numbers
have
2 tenths,
so I so
need
compare
Comparing
0.207 0.207
to a)
0.26,
student
might
think,
“Both
numbers
havec)
2 0.03
tenths,
I need
toI compare
the the the
hundredths.
second
number
has
6 hundredths
and
the first
number
has
no hundredths
so second
the second
hundredths.
The The
second
number
6
hundredths
andfirst
thenumber
first
number
hashundredths
no hundredths
so second
the
hundredths.
The second
number
has
6has
hundredths
and the
has
no
so the
number
must
be
larger.
Another
student
might
think
while
writing
fractions,
I know
is 207
number
must
be larger.
Another
student
might
think
while
writing
fractions,
I know
that that
0.207
is 207
number
must
be
larger.
Another
student
might
think
while
writing
fractions,
I know
that
0.207
is 0.207
207
and
26 hundredths
) Ibut
also
thousandths
(and
and) is
0.26
is
26ishundredths
may
write
canI can
also
think
of it of it
thousandths
(andwrite
may may
write
) write
and )0.26
260.26
hundredths
(and (and
may(and
writemay
)write
but )I but
can
also
think
of itthink
thousandths
(and may
The
Framework
adopted
by California
thenumbers;
California
State
Board
ofin
Education
on November
The Mathematics
Framework
was was
adopted
the
Board
of
Education
onfrom
November
4. Mathematics
Gather
blocks
to represent
each
ofCalifornia
these
then
put
them
order
The Mathematics
Framework
was
adopted
by
theby
StateState
Board
of
Education
on
November
6, 6, 6,
2013.
The
Mathematics
Framework
has
not
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edited
for
publication.
The smallest
Mathematics
Framework
hasbeen
not
been
edited
for publication.
2013. 2013.
The Mathematics
Framework
has
not
edited
for 0.9,
publication.
to largest.
0.71,
0.49,
0.08,
0.6, 0.05
_______ < _______ < _______ < _______ < _______ < _______
!
IMP Activity: Base 10 Blocks and Decimal Place Value
1
PVD S19
5. Remember that multiplying by a whole number can mean to make groups. Multiplying
0.03 by 10 can be represented by 10 groups of 0.03. Gather blocks so that you have 10
groups of 0.03. Draw them below and count what you have. Complete the equation
below.
0.03 × 10 = _________
6. Let’s see what happens when we continue multiplying by 10 again. Multiplying 0.3
by 10 can be represented by 10 groups of 0.3. Gather blocks so that you have 10
groups of 0.3. Draw them below and count what you have. Complete the equation
below.
0.3 × 10 = _________
7. What would have happened if we had multiplied 0.02 by 10 and 0.2 by 10, instead of
using 0.03 and 0.3? Write equations like the ones above and explain with a sentence.
___________________
_________________________________________
___________________
_________________________________________
8. Write four more equations of decimal numbers in the tenths or hundredths being
multiplied by 10. Use the patterns you see to complete the equations or build them
with your blocks.
___________________________
___________________________
___________________________
___________________________
9. Discus with your class and write down conclusions about what patterns you notice
when multiplying decimal numbers by 10.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
!
IMP Activity: Base 10 Blocks and Decimal Place Value
2
PVD S20
!
10. Remember that dividing by 10 or multiplying by !" can mean to split something up
into 10 equal groups, and then keeping just one of the groups. Dividing 2 by 10 can be
represented by splitting 2 flats into 10 equal groups and then keeping one of those
groups. Gather blocks so that you have 2 flats and then divide that by 10. Draw the
results and complete the equivalent equations below.
2 ÷ 10 = _________ or 2 × !
!"
= _________
11. Let’s see what happens when we continue dividing by 10 again. Dividing 0.2 by 10
can be represented by splitting 2 rods into 10 equal groups and keeping one of those
groups. Use the 2 rods you have from above and divide them by 10. Draw the results
and complete the equivalent equations below.
0.2 ÷ 10 = _________ or 0.2 × !
!"
= _________
12. What would have happened if we had divided 5 by 10 and 0.5 by 10, instead of using
2 and 0.2? Write equations like the ones above and explain with a sentence.
___________________
_________________________________________
___________________
_________________________________________
13. Write four more equations of decimal numbers in the tenths or hundredths being
multiplied by 10. Use the patterns you see to complete the equations or build them
with your blocks.
___________________________
___________________________
___________________________
___________________________
14. Discus with your class and write down conclusions about what patterns you notice
!
when dividing decimal numbers by 10 or multiplying by !".
_______________________________________________________________________
_______________________________________________________________________
!
IMP Activity: Base 10 Blocks and Decimal Place Value
3
PVD S21
Name _____________________________________ Date ____________ Period ___________
Block Dragon Your goal is to be the Block Dragon with the largest pile of blocks at the end of round 3.
In each round you will roll a die to get the digits of a decimal number. After each roll you
must decide where to put the number from the die. Two rolls will be used and one won’t.
Round 1
Roll a single die. Write the digit in one of the places below. Then roll again. Write the
digit. Roll a third time and write the digit.
Who got more in Round 1?
0. ___ ___
Unused number ___
You cannot move a digit after you write it.
Gather Base 10 blocks for your pile.
0. ___ ___
me
0. ___ ___
my opponent
Use <, > or = to compare.
Round 2
Roll a single die. Write the digit in one of the places below. Then roll again. Write the
digit. Roll a third time and write the digit.
Who got more in Round 2?
0. ___ ___
Unused number ___
You cannot move a digit after you write it.
Gather Base 10 blocks for your pile.
0. ___ ___
me
0. ___ ___
my opponent
Use <, > or = to compare.
Round 3
Roll a single die. Write the digit in one of the places below. Then roll again. Write the
digit. Roll a third time and write the digit.
Who got more in Round 3?
0. ___ ___
Unused number ___
You cannot move a digit after you write it.
Gather Base 10 blocks for your pile.
0. ___ ___
me
0. ___ ___
my opponent
Use <, > or = to compare.
Who is the Block Dragon?
Count the total amount you have after 3 rounds. Count your opponent’s pile also.
Record below the totals and the winner. Remember to trade 10 hundredths for 1 tenth
and to trade 10 tenths for a whole. Use <, > or = to compare the totals.
___ . ___ ___
___ . ___ ___
my total
my opponent’s total
!
IMP Activity: Block Dragon
___________ is the Block Dragon
1
PVD S22
Name _____________________________________ Date ____________ Period ___________
Equivalent Decimal Expressions Page 1 A
B
Standard
Decimal
Words
Fraction
0.42
forty-two
hundredths
42
100
Fraction
Product
42 × 1
100
Expanded Words
Expanded
Fraction
Expanded Fraction
Product
0.4 + 0.02
four tenths and
two hundredths
4
2
+
10 100
4 ×
1
1
+ 2 ×
10
100
5 ×
1
1
+ 4 ×
10
100
seventy-three
hundredths
C
0.1 + 0.09
90 × D
E
Expanded
Decimal
1
100
36
100
F
0
9
+
10 100
G
!
IMP Activity: Equivalent Decimal Expressions
1
PVD S23
Name _____________________________________ Date ____________ Period ___________
Equivalent Decimal Expressions Page 2 H
Standard
Decimal
Words
Fraction
0.402
four hundred
two thousandths
402
1000
Fraction
Product
402 × 1
1000
Expanded Words
Expanded
Fraction
Expanded Fraction
Product
0.4 + 0.002
four tenths and
two thousandths
4
2
+
10 1000
4 ×
1
1
+ 2 ×
10
1000
fifty-six
thousandths
I
J
0.01 + 0.004
8 × K
1
100
6
1000
L
9 ×
M
N
Expanded
Decimal
1
1
+ 7 ×
10
100
0.528
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IMP Activity: Equivalent Decimal Expressions
2
PVD S24
0.42 0.4 + 0.02 42
100
4
2
+
10 100
1
1
4 × + 2 × 10
100
1
42 × 100
𝑓𝑜𝑟𝑡𝑦 − 𝑡𝑤𝑜 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠 𝑓𝑜𝑢𝑟 𝑡𝑒𝑛𝑡ℎ𝑠 𝑎𝑛𝑑 𝑡𝑤𝑜 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠 EQUIVALENT DECIMALS SORT MATCH PVD S25
0.024 24
1000
0.02 + 0.004 2
4
+
100 1000
1
24 × 1000
1
1
2 × + 4 × 100
1000
𝑡𝑤𝑒𝑛𝑡𝑦 − 𝑓𝑜𝑢𝑟 𝑡ℎ𝑜𝑢𝑠𝑎𝑛𝑑𝑡ℎ𝑠 𝑡𝑤𝑜 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠 𝑎𝑛𝑑 𝑓𝑜𝑢𝑟 𝑡ℎ𝑜𝑢𝑠𝑎𝑛𝑑𝑡ℎ𝑠 EQUIVALENT DECIMALS SORT MATCH PVD S26
0.60 0.6 + 0.00 60
100
6
0
+
10 100
1
1
6 × + 0 × 10
100
1
60 × 100
𝑠𝑖𝑥𝑡𝑦 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠 𝑠𝑖𝑥 𝑡𝑒𝑛𝑡ℎ𝑠 𝑎𝑛𝑑 𝑧𝑒𝑟𝑜 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠 EQUIVALENT DECIMALS SORT MATCH PVD S27
0.03 0.0 + 0.03 3
100
0
3
+
10 100
1
3 × 100
𝑡ℎ𝑟𝑒𝑒 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠 1
1
0 × + 3 × 10
100
𝑧𝑒𝑟𝑜 𝑡𝑒𝑛𝑡ℎ𝑠 𝑎𝑛𝑑 𝑡ℎ𝑟𝑒𝑒 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠 EQUIVALENT DECIMALS SORT MATCH PVD S28
Name _____________________________________ Date ____________ Period ___________
Formative Assessment
Think about the numbers 0.308 and 0.324. Explain clearly using any of the ideas you
have learned so far and complete sentences which of these two numbers is larger.
Give at least two different explanations for how you can tell which is larger. Include
drawings with your explanations when it is useful.
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
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IMP Activity: Formative Assessment
1
PVD S29
Name _____________________________________ Date ____________ Period ___________
Patterns in Place Value Today you will use a calculator to do some multiplication and division. With the
calculator doing the calculating, you can put your attention to looking for patterns
and making predictions. The patterns you will see today are among the most
important of our number system.
Multiplying by 10
1. Begin with the number 37 and multiply it by 10. Write the result in the next box
to the right. Multiply this number by 10 to get the third number in the pattern.
Continue to fill the five boxes.
× 10
× 10
× 10
× 10
37
2. Look at the five numbers. What patterns do you see?
__________________________________________________________________
__________________________________________________________________
3. Look closely at where the digit “7” is in each number. What is happening to the
7 and why is that happening?
__________________________________________________________________
__________________________________________________________________
4. Look closely at where the digit “3” is in each number. What is happening to the
3 and why is that happening?
__________________________________________________________________
__________________________________________________________________
!
IMP Activity: Patterns in Place Value
1
PVD S30
Let’s expand the numbers from the pattern above to see patterns more clearly and
to introduce a new way of writing repeated multiplication.
5. Complete the bottom two boxes of the 2nd and 3rd columns.
370
37×10 37×10
3,700
37×10×10
37×100
37,000
370,000
In the final row, did it make you tired to write 37×10×10×10×10? And this
would just get worse continuing to the millions and billions. Mathematicians have
designed a short-cut way of writing repeated multiplication. This last number in the
chart above can be written:
37×10!
The little number “4” to the top right of the 10 is called an exponent and it tells you
how many times the 10 appears as a factor. Put this number, 37×10! , in the
bottom right box of the chart above. Fill in the remaining three boxes in the last
column using exponents that are 1, 2 and 3.
6. Without using your calculator, explain what 37×10! means and what the
number would look like.
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__________________________________________________________________
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!
IMP Activity: Patterns in Place Value
2
PVD S31
Dividing by 10
7. This time we will start with a large number and divide by 10 again and again.
Use your calculator to fill in the numbers in this pattern.
÷ 10
÷ 10
÷ 10
÷ 10
÷ 10
÷ 10
52,000
8. Look at the seven numbers. What patterns do you see?
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__________________________________________________________________
9. Look closely at where the digit “2” is in each number. What is happening to the
2 and why is that happening?
__________________________________________________________________
__________________________________________________________________
10. Look closely at where the digit “5” is in each number. What is happening to
the 5 and why is that happening?
__________________________________________________________________
__________________________________________________________________
11. How do the effects of dividing by 10 compare to the effects of multiplying by
10?
__________________________________________________________________
__________________________________________________________________
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IMP Activity: Patterns in Place Value
3
PVD S32
12. Let’s expand the numbers from the pattern above to see patterns more clearly
!
and to continue using exponents. The chart below shows multiplying by
!"
repeatedly. Is this equivalent to what you did with your calculator? Explain.
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__________________________________________________________________
13. Complete the bottom three boxes of each column.
5,200
520
52,000×
52,000×
1
10
1
1
×
10 10
1
10
1
52,000×
100
52,000×
1
10!
1
52,000× !
10
52,000×
52
5.2
0.52
!
14. Explain why 52,000 × !"! equals 52.
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__________________________________________________________________
Exponents can be used to show repeated multiplication. They are written as
smaller numbers slightly above and to the right of the number being repeated.
10! = 10 × 10 = 100
10! = 10 × 10 × 10 = 1,000
10! = 10 × 10 × 10 × 10 = 10,000
!
IMP Activity: Patterns in Place Value
4
PVD S33
Here are some questions to check your understanding.
15. How many times larger is 75 than 0.75? Show this with a multiplication
equation.
16. Write two equations, one with multiplication and one with division, that use
the number 3,000 and the number 3.
17. Rewrite your equations from Question 16 using an exponent.
18. Write two equations, one with multiplication and one with division, that use
the number 0.8 and the number 0.008.
19. When a number is multiplied by 10! the result is 730. What was the number?
20. In the number below, how many times larger is the amount represented by the
first digit 5 than the amount represented by second digit 5?
35,452
!
IMP Activity: Patterns in Place Value
5
PVD S34
Team A Team A 750 3 × 10 Round 1 Round 2 Team A !
2 × 10 Round 1 !
Team A 1090 Round 2 HOW FAR NUMBER CARDS PVD S35
Team A Team A !
2 × 10
528 Round 3
Round 4 Team A Team A 1855 Round 3 !
!
10 + 10 Round 4 HOW FAR NUMBER CARDS PVD S36
Team B Team B 750 3 × 10 Round 1 Round 2 Team B !
2 × 10 Round 1 !
Team B 1090 Round 2 HOW FAR NUMBER CARDS PVD S37
Team B Team B !
2 × 10
528 Round 3
Round 4 Team B Team B 1855 Round 3 !
!
10 + 10 Round 4 HOW FAR NUMBER CARDS PVD S38
Team C Team C 750 3 × 10 Round 1 Round 2 Team C !
2 × 10 Round 1 !
Team C 1090 Round 2 HOW FAR NUMBER CARDS PVD S39
Team C Team C !
2 × 10
528 Round 3
Round 4 Team C Team C 1855 Round 3 !
!
10 + 10 Round 4 HOW FAR NUMBER CARDS PVD S40
Team D Team D 750 3 × 10 Round 1 Round 2 Team D !
2 × 10 Round 1 !
Team D 1090 Round 2 HOW FAR NUMBER CARDS PVD S41
Team D Team D !
2 × 10
528 Round 3
Round 4 Team D Team D 1855 Round 3 !
!
10 + 10 Round 4 HOW FAR NUMBER CARDS PVD S42
Team E Team E 750 3 × 10 Round 1 Round 2 Team E !
2 × 10 Round 1 !
Team E 1090 Round 2 HOW FAR NUMBER CARDS PVD S43
Team E !
2 × 10
528 Round 3
Round 4 Team E Team E 1855 Round 3 Team E !
!
10 + 10 Round 4 HOW FAR NUMBER CARDS PVD S44
Team F Team F 750 3 × 10 Round 1 Round 2 Team F !
2 × 10 Round 1 !
Team F 1090 Round 2 HOW FAR NUMBER CARDS PVD S45
Team F Team F !
2 × 10
528 Round 3
Round 4 Team F Team F 1855 Round 3 !
!
10 + 10 Round 4 HOW FAR NUMBER CARDS PVD S46
Team G Team G 750 3 × 10 Round 1 Round 2 Team G !
2 × 10 Round 1 !
Team G 1090 Round 2 HOW FAR NUMBER CARDS PVD S47
Team G Team G !
2 × 10
528 Round 3
Round 4 Team G Team G 1855 Round 3 !
!
10 + 10 Round 4 HOW FAR NUMBER CARDS PVD S48
Team H Team H 750 3 × 10 Round 1 Round 2 Team H !
2 × 10 Round 1 !
Team H 1090 Round 2 HOW FAR NUMBER CARDS PVD S49
Team H Team H !
2 × 10
528 Round 3
Round 4 Team H Team H 1855 Round 3 !
!
10 + 10 Round 4 HOW FAR NUMBER CARDS PVD S50
Team I Team I 750 3 × 10 Round 1 Round 2 Team I !
2 × 10 Round 1 !
Team I 1090 Round 2 HOW FAR NUMBER CARDS PVD S51
Team I Team I !
2 × 10
528 Round 3
Round 4 Team I Team I 1855 Round 3 !
!
10 + 10 Round 4 HOW FAR NUMBER CARDS PVD S52
Team J Team J 750 3 × 10 Round 1 Round 2 Team J !
2 × 10 Round 1 !
Team J 1090 Round 2 HOW FAR NUMBER CARDS PVD S53
Team J !
2 × 10
528 Round 3
Round 4 Team J Team J 1855 Round 3 Team J !
!
10 + 10 Round 4 HOW FAR NUMBER CARDS PVD S54
Team A 0.3 Team A Seven hundred five thousandths Round 1 Round 2 Team A 6
2
+
10 100
Round 1 Team A 0.4 + 0.06 Round 2 IN BETWEEN NUMBER CARDS PVD S55
Team A Two tenths and six hundredths Round 3
Team A Team A 0.582 Round 3 1
8 × 10
Round 4 Team A 0.091 Round 4 IN BETWEEN NUMBER CARDS PVD S56
Team B 0.3 Team B Seven hundred five thousandths Round 1 Round 2 Team B 6
2
+
10 100
Round 1 Team B 0.4 + 0.06 Round 2 IN BETWEEN NUMBER CARDS PVD S57
Team B Two tenths and six hundredths Round 3
Team B Team B 0.582 Round 3 1
8 × 10
Round 4 Team B 0.091 Round 4 IN BETWEEN NUMBER CARDS PVD S58
Team C 0.3 Team C Seven hundred five thousandths Round 1 Round 2 Team C 6
2
+
10 100
Round 1 Team C 0.4 + 0.06 Round 2 IN BETWEEN NUMBER CARDS PVD S59
Team C Two tenths and six hundredths Round 3
Team C Team C 0.582 Round 3 1
8 × 10
Round 4 Team C 0.091 Round 4 IN BETWEEN NUMBER CARDS PVD S60
Team D 0.3 Team D Seven hundred five thousandths Round 1 Round 2 Team D 6
2
+
10 100
Round 1 Team D 0.4 + 0.06 Round 2 IN BETWEEN NUMBER CARDS PVD S61
Team D Two tenths and six hundredths Round 3
Team D Team D 0.582 Round 3 1
8 × 10
Round 4 Team D 0.091 Round 4 IN BETWEEN NUMBER CARDS PVD S62
Team E 0.3 Team E Seven hundred five thousandths Round 1 Round 2 Team E 6
2
+
10 100
Round 1 Team E 0.4 + 0.06 Round 2 IN BETWEEN NUMBER CARDS PVD S63
Team E Two tenths and six hundredths Round 3
Team E 0.582 Round 3 Team E 1
8 × 10
Round 4 Team E 0.091 Round 4 IN BETWEEN NUMBER CARDS PVD S64
Team F 0.3 Team F Seven hundred five thousandths Round 1 Round 2 Team F 6
2
+
10 100
Round 1 Team F 0.4 + 0.06 Round 2 IN BETWEEN NUMBER CARDS PVD S65
Team F Two tenths and six hundredths Round 3
Team F Team F 0.582 Round 3 1
8 × 10
Round 4 Team F 0.091 Round 4 IN BETWEEN NUMBER CARDS PVD S66
Team G 0.3 Team G Seven hundred five thousandths Round 1 Round 2 Team G 6
2
+
10 100
Round 1 Team G 0.4 + 0.06 Round 2 IN BETWEEN NUMBER CARDS PVD S67
Team G Two tenths and six hundredths Round 3
Team G Team G 0.582 Round 3 1
8 × 10
Round 4 Team G 0.091 Round 4 IN BETWEEN NUMBER CARDS PVD S68
Team H 0.3 Team H Seven hundred five thousandths Round 1 Round 2 Team H 6
2
+
10 100
Round 1 Team H 0.4 + 0.06 Round 2 IN BETWEEN NUMBER CARDS PVD S69
Team H Two tenths and six hundredths Round 3
Team H Team H 0.582 Round 3 1
8 × 10
Round 4 Team H 0.091 Round 4 IN BETWEEN NUMBER CARDS PVD S70
Team I 0.3 Team I Seven hundred five thousandths Round 1 Round 2 Team I 6
2
+
10 100
Round 1 Team I 0.4 + 0.06 Round 2 IN BETWEEN NUMBER CARDS PVD S71
Team I Two tenths and six hundredths Round 3
Team I Team I 0.582 Round 3 1
8 × 10
Round 4 Team I 0.091 Round 4 IN BETWEEN NUMBER CARDS PVD S72
Team J 0.3 Team J Seven hundred five thousandths Round 1 Round 2 Team J 6
2
+
10 100
Round 1 Team J 0.4 + 0.06 Round 2 IN BETWEEN NUMBER CARDS PVD S73
Team J Two tenths and six hundredths Round 3
Team J 0.582 Round 3 Team J 1
8 × 10
Round 4 Team J 0.091 Round 4 IN BETWEEN NUMBER CARDS PVD S74