Number Corner, Grade 4 • 67
11
18
10
17
Calendar Grid
4
Monday
3
Sunday
12
5
Tuesday
13
6
Wednesday
14
7
Thursday
9
8
16
2
Saturday
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Day
1
4
1
2
3
4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1
1 41
1 21
1 43
2
2 41
2 21
2 43
3
3 41
3 21
3 43
4
4 21
4 21
Quarts
Cups
1
8
4 = 1
4
16
5
16
6
16
7
16
8 = 1
2
16
9
16
10
16
11
16
12
16
13
16
14
16
15
16
1
16
2
16
3
16
Gallons
1
1 161
1 162 = 1
Cups, Quarts & Gallons Record Sheet
Calendar Collector
Multiples of 4
1
15
Friday
Multiples of 3
October
Multiples of 2
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
Number Line
0 1
October
October Calendar Collector
CALENDAR COLLECTOR
A Cup a Day
Overview
You’ll need
The class pours a cup of tinted water each
day into a quart container. When 4 quarts
are filled, the water is poured into a gallon
container. The class keeps a chart to show
the growing collection of cups, quarts,
and gallons.
H Cups, Quarts & Gallons Record Sheet,
pages 1 and 2 (Blacklines NC 2.1 and
2.2, 1 copy of each trimmed and glued
together and posted on the display
board by the second week)
Frequency
H A Cup a Day, pages 1 and 2 (Number
Corner Student Book, pages 20 and 21)
One day per week. Have a student volunteer add a cup of water to the collection
for every day of the month.
H 1-cup liquid measuring cup
Skills & Concepts
H measuring capacity with accuracy
H 1 plastic gallon milk jug (see Advance
Preparation)
H using U.S. customary measures of
capacity to conceptualize fractions
H clean cup, quart, and gallon containers
brought in by students (optional)
H exploring equivalent fractions
H small bottle of food coloring
H carrying out simple unit conversions
within the U.S. customary system
H masking tape
H making realistic estimates and measurements using cups, quarts, and
gallons and selecting the unit most
appropriate for a given situation
H multiplying and dividing by 4 or 16
H representing and analyzing patterns
and functions using a table
H funnel
H 4 quart capacity containers
H permanent black felt tip marker
Advance Preparation Bring in your own
plastic gallon milk jug. Run a strip of masking tape up the side of the gallon jug and
each quart container so that you can mark
the level of the water each day.
Note If you want students to fill in their
own copies of the Calendar Collector
Record Sheet, run a double class set of
Blackline NC 2.1, and have students store
the two sheets in their math binders.
Number Corner, Grade 4 • 81
October
Calendar Collector A Cup a Day (cont.)
Week 1 Introducing the Calendar Collector
Explain that this month, students will collect a cup of water a day including weekends. Pose a few brief questions to find out what students already
know and think about cups. How big is a cup? Does a cup always mean the
same thing? If someone offered them a cup of juice on a hot day, would it be
enough? How many cups would be enough to quench a big thirst? How many
cups of milk do they drink each day? Who uses cup measures?
Next, hold up the 1-cup liquid measuring cup. When would someone use a
cup like this to measure liquid with accuracy? When your fourth graders look
at the 1-cup measure and think about their cups and glasses at home, do they
think they’re probably drinking more or less than an actual cup of liquid
each time they have a glass of juice or milk?
Then show students one of the quart containers and explain that each day,
someone will add a very carefully measured cup of water to this quart container and mark the level on the masking tape on the side of the container to
show how much has been added. How many days do they think it will take
to fill this container? (Even though many students will have measured with
cups and quarts in earlier grades, their estimates will likely vary quite a bit.)
Don’t tell them that a quart holds 4 cups: they will come to understand this
fact much more powerfully through the experience of measuring and pouring themselves.
Next demonstrate how to measure a single cup of water into the quart container. Show students how to use the measuring cup with accuracy, add a
couple of drops of food coloring so the water will be visible in the container,
use the funnel as you pour the water into the container, and mark the tape
on the side of the quart to show that 1 cup has been added.
Now that they can see how much of the container is filled by 1 cup, ask if
they would like to adjust their estimates regarding the total number of cups
82 • Number Corner, Grade 4
October
Calendar Collector A Cup a Day (cont.)
the container will hold. Many will see that the container is about one-quarter full, and some may remember that there are 4 cups in a quart. With students’ help, measure and pour as many cups into the first quart container as
the number of days that have passed so far this month. You will need to use a
second quart container if more than 4 days have gone by. Mark the new level
on the container as you pour in each cup, and ask your students to identify
how much of the container has been filled each time. This is a wonderful opportunity to discuss fractions of a quart —1⁄4, 1⁄2, 3⁄4, and 4⁄4, or 1 full quart.
After pouring the cups, you might pose some of the following questions,
many of which can be repeated several times over the course of the month.
• How many cups have we collected so far?
• How many quarts is that? (Again, this is a great opportunity to use the
language of fractions; there are many times when the class will have collected enough water to fill a whole number of quarts as well a fractional
part of a quart. On the 5th of October, for instance, 5 cups will have been
collected, which is equal to 1 and 1⁄4 quarts.)
• How long will it be before we’ve collected enough to fill a quart? 2 quarts?
3 quarts? A gallon?
• How many quarts do you think we will have collected by the end of the
month? How many gallons?
• Name a few things around our classroom that would hold about a cup of
water. Can you spot some containers that would hold about a quart of water? A gallon? 5 gallons?
Week 2 Introducing the Record Sheet & Discussing Fourths
Have a student helper add a cup to the quart container each day. On Mondays, you may need to remind the helper to add 3 cups, one for that day and
2 for Saturday and Sunday. To help students make the connection between
these units of measure and their daily lives, you might ask them to bring in
containers from home that hold a cup, a quart, or a gallon, such as milk and
juice cartons and jugs, pickle jars, pop bottles, and so on.
The second time you do this workout together as a class, introduce the Cups,
Quarts & Gallons Record Sheet to help the class keep track of how many cups
they have collected. The sheet will also prompt them to group and divide
by fours, and to think about fourths and possibly sixteenths. By the second
week, students will have collected about 7 or 8 cups (filling 13⁄4 or 2 quarts),
perhaps more. Ask students to help you fill in the data for each day that has
passed so far this month.
After the data has been entered, ask students to share observations about the
numbers on the record sheet. As they discover the patterns in the table the
Number Corner, Grade 4 • 83
October
Calendar Collector A Cup a Day (cont.)
first day it’s posted, students may begin to make conjectures and generalizations about fractions and about converting between cups, quarts, and gallons.
If debate ensues, keep your questioning open rather than stepping in with explanations. Students may not resolve some questions, even by the end of the
month, but they will have many more experiences with fractions in various
contexts this year.
Cups, Quarts & Gallons Record Sheet
Day
Cups
Quarts
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
4
1
2
3
4
Gallons
1
1 41
1 21
1 43
Antoine The numbers in the cups go 1, 2, 3, 4, 5, and the numbers in
the quarts go 1⁄4, 1⁄2, 3⁄4, 1, 1 1⁄4, 1 1⁄2, 1 3⁄4. Most of the quart numbers
have fractions, but the cups don’t.
Teacher Why does it work like that?
Students The cups are just going one by one—we add a whole cup every day.
But the cups don’t always fill the quarts. Most of the time we just have a
quart and then part of a quart.
Some of the numbers match, like 1 cup is the same as 1 quarter, and 3
cups is the same as 3 quarters. It would be cool if the chart said 1⁄4, 2⁄4,
3⁄4, 4 ⁄4, so the numbers would match.
glue or tape sheet 2 here
Teacher Can we do that?
Students It seems like 2⁄4 is the same as a half. I mean, you can see it
on the quart. When we’ve put in 2 cups, it’s like 2⁄4, but half the quart is
filled.
I don’t think we can do it. There’s no such thing as 4 ⁄4.
Week 3 Introducing Gallons & Discussing Sixteenths
After the class has filled all 4 quart containers, they will pour the water into
the gallon container. The following day, they’ll begin to fill one of the nowempty quart containers. In the days that follow, students may fill in some or
all of the values in the gallon column on the record sheet, which will involve
84 • Number Corner, Grade 4
October
Calendar Collector A Cup a Day (cont.)
sixteenths, although some students may observe that 4⁄16 is the same as 1⁄4 or
8⁄16 is the same as 1⁄2 because 1 quart fills a quarter of the gallon container,
and 2 quarts fills a half. If students are not ready to work with sixteenths,
they can simply enter a 1 in the gallon column on the 16th and leave it blank
for the rest of the month. By the 31st day of the month, they will have filled 1
gallon, 3 and 3⁄4 quarts.
Cups, Quarts & Gallons Record Sheet
Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Cups
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Quarts
Gallons
1
4
1
2
3
4
1
16
2
16
3
16
1
1 41
1 21
1 43
2
2 41
2 21
2 43
3
3 41
3 21
3 43
4
4 21
4 21
4 = 1
4
16
5
16
6
16
7
16
8 = 1
2
16
9
16
10
16
11
16
12
16
13
16
14
16
15
16
1
1 161
1 162 = 1
1
8
Week 4 Making Conversions Independently
Toward the very end of the month, have students work individually or in
pairs to complete pages 20 and 21 in place of a whole group discussion.
Students’ work on the sheet will provide you with a mini-assessment of their
current proficiency at multiplying and dividing by 4’s; their sense of unit size
when it comes to cups, quarts, and gallons; and their strategies for solving
story problems that require more than one step.
As students fill in the table on page 20, many will skip count rather than
thinking in terms of multiplication. Don’t be alarmed if this is the case, as
the conversion problems will force them to think multiplicatively. If you feel
they need reinforcement with this way of thinking, make time for them to
discuss problems 3 and 5.
Number Corner, Grade 4 • 85
Number Corner Student Book
NAME
DATE
A Cup a Day page 1 of 2
CALENDAR COLLECTOR
1
Fill in the missing values on the
table below. The 3 dots in some boxes
mean that some numbers have been
skipped. You don’t have to put anything in those boxes.
Cups
Quarts
4
1
8
2
2
How many cups are there in 400
quarts? How do you know?
3
If you have 200 cups, how many
quarts can you fill? How do you know?
3
16
4
20
4
Would you use cups, quarts, or gallons to measure the amount of water
you use to take a shower? Why?
32
9
40
10
•
•
•
•
•
•
64
•
•
•
•
•
•
100
20
Number Corner
© The Math Learning Center
Number Corner Student Book
NAME
DATE
A Cup a Day page 2 of 2
CALENDAR COLLECTOR
5
Marco is having a party. He needs to make 3 big pitchers full of lemonade.
Each pitcher takes 12 cups of water and 2 cups of lemon juice. How many quarts
of water will he need? How many quarts of lemon juice will he need? Use numbers, pictures, and words to show your thinking.
a Marco will need _____________ quarts of water.
b Marco will need _____________ quarts of lemon juice.
© The Math Learning Center
Number Corner
21
October Blackline NC 2.1 Run 1 copy, trim, and attach to Blackline NC 2.2. Post it on your calendar display.
Cups, Quarts & Gallons Record Sheet page 1 of 2
Cups, Quarts & Gallons Record Sheet
Day
Cups
Quarts
Gallons
Glue or tape page 2 here.
© The Math Learning Center
Number Corner
October Blackline NC 2.2 Run 1 copy, trim, and attach to Blackline NC 2.1. Post it on your calendar display.
Cups, Quarts & Gallons Record Sheet page 2 of 2
Number Corner
© The Math Learning Center
Number Corner, Grade 4 • 219
★
★
★
★
★
★
★
Multiples of 3
1
2
0
2
4
2
3
1
3
4
3
5
3
2
4
4
2
4
5
4
4
1
0
1
40 24 81 32 40
45 42 25 56 32
56 28 48 30 36
36
54
42
35
72
35
Calendar Collector
16
15
14
13
12
11
10
9
54
49
63
20
81
45
54
54
81
20
63
2
3
1
1
3
63
25
56
49
45
72
28
81
25
64
49
1
Number of
Number of
Odd Products Even Products
32
42
20
36
45
45
25
30
20
20
72
36
72
40
30
8
7
6
5
4
3
64
48
16
48
35
72
40
20
32
54
48
48
42 20 20 30 63
1
2
Products
Day
Roll & Multiply Record Sheet
Multiples of 2
13
6
40
19
Sunday
Multiples of 4
14
7
43
22
Monday
15
8
1
46
25
4
Tuesday
9
2
28
7
Wednesday
10
3
31
10
Thursday
11
4
34
13
12
5
37
16
Saturday
★ Multiples of 7
Friday
Multiples of 6
February
Multiples of 5
13
16
19
22
25
28
31
34
37
40
43
46
4
5
6
7
8
9
10
11
12
13
14
15
Calendar Grid
4
7
10
3 x 15 = 45. Add one more and it’s 46.
If you made a picture, it would be 3 legs of 14 and
1 more. That’s 43.
Even number.
Odd number.
This is the same thing. It’s 11 + 11 + 11 + 1 more to
get to 34.
It’s the same as 10 + 10 + 10, then add 1 more.
It’s an even number.
We were right. It added 17, because 8 + 17 = 25.
It added 15 today. Tomorrow it will add 17. So
the Out number will be 25, because 8 + 17 = 25.
Every new number on the Out side is 3 more
than the one before.
Tomorrow it will be 10, and the day after,
it will be 22. It’s always 3 more.
It goes up by 1 on the In number and by 3
on the Out number.
It added 7 this time.
Now it added 5, not 3.
It added 3 today. Maybe it’ll do that
every time.
Observations & Predictions
February Calendar Record Sheet
Output Number
1
2
3
Input Number
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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Number Line
0 1
★
February
February Calendar Collector
CALENDAR COLLECTOR
Roll & Multiply
Overview
You’ll need
This month, students collect data from
repeated trials of a probability experiment
in which they roll two dice marked 4–9
and multiply the two numbers. Before
conducting the experiment, they predict
how likely it is that a given product will be
odd and how likely it is to be even. In the
middle of the month, after conducting 5
trials every day, students revise their predictions and then continue making 5 trials
a day through the end of the month.
H Roll & Multiply Data Chart (Overhead
NC 6.5)
Frequency
H Roll & Multiply Record Sheet, pages
1 and 2 (Blacklines NC 6.3 and 6.4, 1
copy each, see Advance Preparation)
H Roll & Multiply Data Chart (Number
Corner Student Book, page 63)
H Thinking about Roll & Multiply (Number
Corner Student Book, page 65)
H One More Look at Roll & Multiply
(Number Corner Student Book, page 69)
Update the data daily, and share observations and predictions about the data as a
whole group once or twice a week.
H 2 dice marked 4–9
Skills & Concepts
H yellow highlighter marker
H using multiplication facts through 9 × 9
with fluency
H overhead pens
H predicting the probability of various
outcomes or events
Advance Preparation Run 1 copy each
of Blacklines NC 6.3 and 6.4. Trim and then
glue or tape them together to form one
long chart. Post on your calendar display
board before conducting your first Calendar Collector Workout this month. You
can also run a class set if you want each
student to keep a record.
H representing all possible outcomes for
a simple probability situation
H conducting a probability experiment
H constructing, reading, and interpreting
bar graphs
H calculators
H black felt-tip marker
Week 1 Introducing Roll & Multiply
Introduce Roll & Multiply by holding up two dice marked 4–9. Ask students
what the chance of getting an odd product would be if you rolled the two dice
and multiplied the numbers that came up. What would be the chance of getting an even product? Is there a better chance of getting one than the other,
or is it equally likely that you’ll get an odd or an even product? What if you
repeated the experiment 100 times? Would you get odd products more often
Number Corner, Grade 4 • 233
February
Calendar Collector Roll & Multiply (cont.)
than even, even more often than odd, or about the same number of odd and
even products? Give students time to consider these questions and discuss
their reasoning in pairs and as a whole group.
Explain that they will conduct 5 trials of this experiment for each day in February, including weekends and holidays. Then ask them to discuss the following questions:
• How many rolls will that be in all? (5 rolls per day × 28 days = 140 rolls
or 5 rolls per day × 29 days = 145 rolls)
• Does the class really need to collect that much data? How much do they
think is enough to determine the chances of rolling an odd or an even
product?
• How many rolls will they need to make today? (The number of rolls will
depend on how many days have passed already this month.)
Finally, have students roll and multiply as you record the products on the
Calendar Collector Record Sheet. You might do this by calling students up
one by one or having them pass the 2 dice from student to student. After recording all of the products, ask students to classify each one as odd or even,
and highlight the odd products with a colored marker. Finally, ask them to
count how many odd and even products they got in each group of 5 trials,
and record those totals on the record sheet. Some students may have difficulty identifying whether these larger numbers are odd or even. If so, take
some time for students to discuss how they can tell. Some may think about
dividing each by 2, while others may see that they can refer to the number
line to see if each number is marked as a multiple of 2. At some point, some
students may also see that if one of the multipliers is even, the product will
also be even.
Roll & Multiply Record Sheet
Day
Number of
Number of
Odd Products Even Products
Products
1
42 20 20 30 63
1
4
2
40 64 72 45 36
1
4
3
4
9 5
6
6
5
9 5
6
7
8
To conclude the first workout, ask students to share observations and pre9
dictions about the likely outcomes of this experiment over the course of the
10
month. Based on the data they collected today, many fourth-graders will sug11
gest that they will get more even than odd products but may not be able to ex12
plain why. Others may be convinced that they are just as likely to roll an even
13
or odd product, because there are 3 odd and 3 even numbers on each die.
14
Glue or tape page 2 here.
234 • Number Corner, Grade 4
February
Calendar Collector Roll & Multiply (cont.)
Week 2 Showing the Data on a Chart & Graph
Before conducting the second Calendar Collector Workout, have a student
helper make sure the record sheet is up to date. Then have another student
or pair of students find the total number of times odd and even products have
been rolled and enter those totals on the overhead Roll & Multiply Data Chart.
Emphasize that this task needs to be done carefully and accurately. You may
want to check students’ work quickly before proceeding with the workout.
Roll & Multiply Record Sheet
Day
February Overhead NC 6.5
Number of
Number of
Odd Products Even Products
Products
1
42 20 20 30 63
1
4
2
40
20
32
54
48
48
36
54
42
35
72
35
1
0
1
4
5
4
3
1
3
2
4
2
72 45 63 54 32
2
3
3
4
5
6
7
8
64
48
16
48
35
72
72
28
81
25
64
49
45
54
54
81
20
63
Roll & Multiply Data Chart
CALENDAR COLLECTOR
Date
Total odd products
2/8
12
Total even products
Total products
28
40
1 Label the axes on the graph so you
can show the data from the chart
above on it.
2
Transfer the data from the chart
above to the bar graph.
3
Based on the data, how would you
describe the chance of getting an odd
number when you roll and multiply?
9
impossible
unlikely
equally likely or unlikely
likely
certain
10
11
12
4
How would you describe the chance
of getting an even number when you
roll and multiply?
13
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
14
Glue or tape page 2 here.
impossible
unlikely
equally likely or unlikely
likely
certain
35
30
25
20
15
10
5
0
Display the Roll & Multiply Data Chart with the totals filled in at the top.
Give students a few minutes to share observations and conjectures about the
data, and then ask them to think about how they can transfer the data to the
bar graph on the overhead. Remind them that they need to label both axes of
the graph and ask them to decide together what those labels should say. When
they have decided, label the axes on the overhead.
Then read questions 2 and 3 together and have students share their current
understandings of the terms impossible, unlikely, equally likely or unlikely, likely,
and certain. Students’ understandings of these terms are often intuitive at this
time of year, and that is fine. Next, have students complete page 63 in their
Student Books, which is identical to the overhead. They will need to fill in the
Number Corner, Grade 4 • 235
February
Calendar Collector Roll & Multiply (cont.)
totals, label the axes, and then fill in the bar graph with the totals and answer
questions 2 and 3.
As they finish working on their sheets, have students meet in pairs to compare their graphs and the ways they described the probability of getting odds
and evens. They’ll discuss these ideas further in the workout next week.
Week 3 Thinking about the Data So Far
Before conducting the third Calendar Collector Workout, have a student
helper make sure the record sheet is up to date. Then have another student
or pair of students find the total number of odd and even products that have
been rolled so far. Ask them to enter those totals on the overhead Roll & Multiply Data Chart. By now, the data should show that even products come up
far more frequently than odd products.
Date
Total odd products
Total even products
Total products
Display the overhead with the totals filled in and give students a couple of
minutes to share observations and conjectures about the data. Then ask them
to complete page 65 in their Student Books.
The multiplication table on the page will help students see that the chances
of rolling an odd product are 9 out of 36 (or 1 out of 4) and the chances of rolling an even product are 27 out of 36 (or 3 out of 4). While fourth graders may
not express these probabilities in numerical form, the sheet will help them
begin to see why they have gotten more even than odd products.
236 • Number Corner, Grade 4
February
Calendar Collector Roll & Multiply (cont.)
Number Corner Student Book
NAME
DATE
Thinking about Roll & Multiply
CALENDAR COLLECTOR
Date
Total odd products
1 What observations can you make
about the data above?
Total even products
Total products
3a
How many products are there altogether on the multiplication table?
b
How many of those products are odd?
c
2
Fill in the missing numbers on this
multiplication table. Then color in the
squares with odd products.
×
4
5
4
16
20
5
20
6
24
7
28
8
32
9
6
7
24
30
54
d What does this tell you about the
Roll & Multiply experiment?
45
48
49
48
45
9
32
35
36
35
8
How many of those products are
even?
56
63
64
63
81
As they finish working on their sheets, have students meet in pairs to compare and discuss their work, especially their responses to the final question.
When everyone is finished, ask them to discuss the last question as a class.
Week 4 Drawing Conclusions about the Data & the Experiment
Before you conduct the last Calendar Collector Workout, record on the Roll &
Multiply Data Chart overhead the total number of times odd and even products have come up so far this month. To open the last workout, display the
overhead with the totals filled in and invite students to share observations.
Then ask them to think about how the total number of odd products relates
to the total number of even products. (It is likely that students will notice
that there are about 3 times as many even products as odd products and that
about a fourth of all the products are odd.) After students have had a chance
to discuss the results as a whole group, have them complete page 69 in their
Student Books.
If there are still a few days remaining in the month, ask students to think
about whether the additional data is likely to alter their conclusions. Have
student helpers continue to collect and record data through the remainder of
the month, and if there is time and high student interest, take a few minutes
at the very end of the month to re-examine the totals one last time.
Number Corner, Grade 4 • 237
February Overhead NC 6.5 (Accompanies Number Corner Student Book pages 63, 65, and 69.)
Roll & Multiply Data Chart
CALENDAR COLLECTOR
Date
Total odd products
1
Label the axes on the graph so you
can show the data from the chart
above on it.
2
Transfer the data from the chart
above to the bar graph.
3
Based on the data, how would you
describe the chance of getting an odd
number when you roll and multiply?
impossible
unlikely
equally likely or unlikely
likely
certain
4
How would you describe the chance
of getting an even number when you
roll and multiply?
Total even products
Total products
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
impossible
unlikely
equally likely or unlikely
likely
certain
35
30
25
20
15
10
5
0
© The Math Learning Center
Number Corner
February Blackline NC 6.3 Run 1 copy, trim, and attach to Blackline NC 6.4. Post it on your calendar display.
Roll & Multiply Record Sheet page 1 of 2
Roll & Multiply Record Sheet
Day
Products
Number of
Number of
Odd Products Even Products
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Glue or tape page 2 here.
© The Math Learning Center
Number Corner
February Blackline NC 6.4 Run 1 copy, trim, and attach to Blackline NC 6.3. Post it on your calendar display.
Roll & Multiply Record Sheet page 2 of 2
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Number Corner
© The Math Learning Center
Number Corner Student Book
NAME
DATE
Roll & Multiply Data Chart
CALENDAR COLLECTOR
Date
Total odd products
1
Label the axes on the graph so you
can show the data from the chart
above on it.
Total even products
Total products
125
120
115
110
2
Transfer the data from the chart
above to the bar graph.
3
Based on the data, how would you
describe the chance of getting an odd
number when you roll and multiply?
impossible
unlikely
equally likely or unlikely
likely
certain
105
100
95
90
85
80
75
70
65
60
55
4
How would you describe the chance
of getting an even number when you
roll and multiply?
impossible
unlikely
equally likely or unlikely
likely
certain
50
45
40
35
30
25
20
15
10
5
0
© The Math Learning Center
Number Corner
63
Number Corner Student Book
NAME
DATE
Thinking about Roll & Multiply
CALENDAR COLLECTOR
Date
Total odd products
1
What observations can you make
about the data above?
Total even products
Total products
3a
How many products are there altogether on the multiplication table?
b
How many of those products are odd?
c
2
Fill in the missing numbers on this
multiplication table. Then color in the
squares with odd products.
×
4
5
6
4
16
20
24
5
20
30
6
24
36
7
28
8
32
9
35
7
35
© The Math Learning Center
What does this tell you about the
Roll & Multiply experiment?
45
48
49
54
9
d
32
48
45
8
How many of those products are
even?
56
63
64
63
81
Number Corner
65
Number Corner Student Book
NAME
DATE
One More Look at Roll & Multiply
CALENDAR COLLECTOR
1
Fill in the chart below with the total number of odd and even products rolled
so far.
Date
Total odd products
Total even products
Total products
2a
Circle the pie graph below that you think comes closest to showing the
results of your experiment so far.
Evens
Evens
Odds
Evens
Odds
b
Odds
Evens
Odds
Explain your choice above.
© The Math Learning Center
Number Corner
69
Number Corner, Grade 4 • 297
4 cm
10
8 cm
5 cm
17
3 cm
6 cm
3 cm
11
4 cm
3 cm
18
3 cm
6 cm
4 cm
3 cm
Isosceles Triangle
2 cm
2 cm
3 cm
2 cm
12
1 cm
1 line of symmetry
4 cm
3 cm
1 cm
5
4
3 cm
Tuesday
Monday
Calendar Grid
1 line of symmetry
3 cm
Isosceles Triangle
5 cm
Regular Pentagon
1 cm
3
Sunday
5 cm
Equilateral Triangle
13
Regular Pentagon
14
Rectangle
Area = 14 square cm
7 cm
Equilateral Triangle
3 cm
7
6
2 cm
Thursday
Wednesday
April
6 cm
2 cm
4 cm
Square
3 cm
2 cm
15
Area = 9 square cm
1
8
Equilateral Triangle
1 cm
Friday
3 cm
4 cm
1 cm
2 cm
9 cm
3 cm
9
7 cm
16
2 cm
2 cm
Rectangle
1 cm
2
Saturday
17
18
14
15
16
9
10
11
12
13
7
8
3+1 +1+2=
Rectangle
11 cm
2 + (2 x 3)+ (2 x 4) =
4+7+9=
8+3+5+3=
(2 x 6) + (2 x 3) + 4 =
Trapezoid
Pentagon
6+2+3+2+4=
2 + 2 + 7 + 7 or ( 2 x 2) + ( 2 x 7) =
22 cm
19 cm
20 cm
17 cm
18 cm
15 cm
16 cm
3 + 3 + 3 + 4 or (3 x 3) + 4 = 13 cm
5 + 5 + 4 or (2 x 5) + 4 = 14 cm
1+2+2 +3+3=
Scalene Triangle
Pentagon
9 cm
10 cm
7 cm
8 cm
5 cm
6 cm
3 cm
Total
3 + 3 + 3 + 3 = 12 or 4 x 3 = 12 cm
Equilateral Triangle 5 x 3 =
Pentagon
Trapezoid
Isosceles Triangle
Pentagon
Square
Equilateral Triangle 3 + 3 + 3 = 9 or 3 x 3 =
2 + 2 + 2 + 2 + 2 = 10 or 5 x 2 =
Trapezoid
Regular Pentagon
3+3+2=
1+1+1+1+1=
Regular Pentagon
3
4
5
6
Isosceles Triangle
(2 x 1) + (2 x 2) =
Rectangle
1 + 1 + 1 = 3 or 3 x 1 =
Equilateral Triangle
2
Calculations
Perimeter
April Calendar Record Sheet
Shape Name
1
Date
1 1
12 12
1
12
Calendar Collector
1 + 1 + 1
3
2
12
1
+ 1 + 1 = 1 3
12 3 +
1
1
23
1 + 1 + 1
6
12 12
+ 1 + 1 = 1
3
3
Teacher’s
Collection
1 + 1 + 1
2
6
6
+ 1 + 1 = 1 5
12 2 + 12
5
1
2 12
1 12
1 + 1 + 1
3
3
6
1
+ 1 + 1 = 1
12
6 12
Class
Collection
1
3
1
3
1
3
1
6
1
2
1
2
Great Fraction Race Game Board
April Overhead NC 8.2
1
1
1
1
1
3
1
1
1
1
3
1
2
1
6
1
6
1
3
1
6
1
12
1 1
12 12
1
6
April
April Computational Fluency
COMPUTATIONAL FLUENCY
Division Capture
Overview
You’ll need
This month’s Computational Fluency Workout provides practice with basic division
facts in the context of a simple but engaging strategy game.
H Introducing Division Capture (Overhead NC 8.4)
Frequency
H Division Capture Game Sheets 1–3
(Number Corner Student Book, pages
86, 88, and 92)
One day per week
Skills & Concepts
H developing efficient strategies for
doing basic division facts
H Division Capture Instructions (Number
Corner Student Book, page 85)
H 1 die numbered 1–6
H 2 dice, one marked 1–6 and the other
4–9, for each pair of students
H half-class set of calculators
H overhead marking pens in two different colors
H colored pencils
In the last two months’ worth of Computational Fluency Workouts, students
will play partner games that provide practice with basic facts. This month
features Division Capture, a game in which partners take turns rolling a die
and using the number that comes up to complete one of 20 division equations
on a grid. Each partner uses a different color to write their numbers on the
grid, and once all the equations are completed, players seek out any equations they completed that fall in a row, either vertically, horizontally, or diagonally. Each player earns a point for any 3 equations in a row and 2 points for
any 4 equations in a row.
Week 1 Introducing Division Capture
Begin the first workout by explaining that this month students will play a new
game called Division Capture. After playing a demonstration game with you
at the overhead today, they will play in pairs for the rest of the month. Then
display the text at the top of the Introducing Division Capture overhead. Read
the game rules out loud to the class, and ask students to choose a pen color
for the class and a different color for you. Fill in the boxes on the overhead to
show which colors you’ll be using. Then take turns with a volunteer rolling
the die to determine whether you or the students will go first.
Number Corner, Grade 4 • 319
April
Computational Fluency Divison Capture (cont.)
Regardless of whether you or the students go first, once the number is rolled,
ask students to study the 20 equations on the grid quietly and then raise
their hands when they have found one or more that will work. Give students
plenty of time so that nearly everyone has a chance to find an equation that
will work, and let them know that there will be more than one equation that
can be completed with this number. (There will be between 2 and 4 equations on the grid that can be completed with any number on the die.) When
students identify the equations that would work with this number, ask them
to explain how they know that the number will make the equation true.
April Overhead NC 8.4
Introducing Division Capture
1
Have each team choose a color and
fill in the boxes below to show what
they are. Then roll the 1–6 die to see
who goes first (high number starts).
2
5
Roll the die and use the number
you get to make one of the equations
below true. Write the number in the
box using your color.
3 6
3
Take turns until all the boxes are
filled. (If you roll a number you can’t
use, you lose that turn.) Try to capture 3 or 4 boxes in a row—across, up
and down, or diagonally. After all the
boxes are filled, circle the places on
the grid where you got 3 or 4 in a row,
and then add up both scores. You get 1
point for every set of 3 in a row and 2
points for every set of 4 in a row.
Students
Teacher
40 ÷
=8
21 ÷
=7
15 ÷
=3
7÷
=7
32 ÷
=8
10 ÷
= 10
20 ÷
=5
24 ÷
=8
35 ÷
=7
18 ÷
=3
14 ÷
=7
16 ÷
=4
36 ÷
=9
18 ÷
=9
25 ÷
=5
24 ÷
=4
30 ÷
=5
18 ÷
=6
36 ÷
=6
24 ÷
= 12
Scoring:
3 in a row—1 point
4 in a row—2 points
Students’ Score
Teacher’s Score
Teacher I rolled a 5. Which equations can I complete by writing a 5 in
the box? I’m going to ask that we all study the game board in silence and
when you see several equations that would work, just raise your hand.
When I see lots of hands, I’ll call on people to share their ideas. …
Sage Five would work in that one in the top row that says, “15 divided
by box equals 3.” Then it would be 15 ÷ 5 = 3, and I know that’s true
because 3 × 5 = 15.
Keith I see another one that would work. If you put the 5 in the very
first box at the top, the sentence would say, “40 ÷ 5 = 8.” I know it works
because 8 × 5 = 40.
320 • Number Corner, Grade 4
April
Computational Fluency Divison Capture (cont.)
Susie You could also use it for 25 ÷ 5 = 5 or 35 ÷ 5 = 7.
Teacher Are there any other places 5 would work as a divisor to complete the equation? No? So there are 4 possibilities. Which one should I
choose?
Students The one in the top corner!
Do 25 ÷ 5 = 5. I like that one.
It doesn’t really matter right now.
You should choose one kind of in the middle so you’ll have a better
chance of getting others that line up with it later. I learned that from
playing tic-tac-toe with my big brother.
Take turns rolling and recording with the class, and have a different student
roll and record for the class each time it is their turn. Continue to give students time to think carefully about their choice of equation, especially toward
the middle of the game when they will need to strategize in order to capture
adjacent equations and block you from capturing adjacent equations. If you or
the student rolls a number that can’t be used, play passes to the other player.
Toward the end of the game, you may have to pass the die back and forth a
Introducing Division Capture
number of times until you or they are able to capture the last few equations.
April Overhead NC 8.4
1 Have each
team choose a color and
3 Take turns
until all the boxes are
When all 20 equations
have
been completed,
ask
a student volunteer to circle
fill in the boxes below to show what
filled. (If you roll a number you can’t
they are. Then roll the 1–6 die to see
use, you lose that turn.) Try to capin their pen colorwho
any
equations captured
by the class that fall 3 or 4 in a row.
goes first (high number starts).
ture 3 or 4 boxes in a row—across, up
and down, or diagonally. After all the
2
Roll
the
die
and
use
the
number
Do the same for yourself using your pen
color,
then
have students use
boxes
are filled,and
circle the
places on
you get to make one of the equations
the grid where you got 3 or 4 in a row,
below
true.
Write
the
number
in
the
the scoring guide at the bottom of the overhead
to scores.
calculate
and then add up both
You get 1 both scores.
box using your color.
Students
point for every set of 3 in a row and 2
points for every set of 4 in a row.
Teacher
40 ÷
5
=8
21 ÷
3
=7
15 ÷
5
=3
7÷
32 ÷
4
=8
10 ÷
1
= 10
20 ÷
4
=5
24 ÷
3
=8
35 ÷
5
=7
18 ÷
6
=3
14 ÷
2
=7
16 ÷
4
=4
36 ÷
4
=9
18 ÷
2
=9
25 ÷
5
=5
24 ÷
6
=4
30 ÷
6
=5
18 ÷
3
=6
36 ÷
6
=6
24 ÷
2
= 12
Scoring:
3 in a row—1 point
4 in a row—2 points
Students’ Score
4
1
=7
Teacher’s Score
4
Students Hey! It turned out to be a tie game!
I thought we were going to lose when Mrs. MacIntosh got 24 ÷ 6 and it
gave her 4 in a row.
We really lucked out when we got 36 ÷ 4 because that gave us 3 in a row
in 2 directions.
Number Corner, Grade 4 • 321
April
Computational Fluency Divison Capture (cont.)
Note Don’t erase this overhead until after the next Computational Fluency Workout. Displaying the completed game board next week will help you review the game
before students play on their own in pairs.
Weeks 2–4 Playing Division Capture
Open the second Computational Fluency Workout by quickly reviewing Division Capture. Display the completed overhead from the first week and have
students read the game instructions on page 85 in their Number Corner Student Books.
Number Corner Student Book
NAME
DATE
Division Capture Instructions
COMPUTATIONAL FLUENCY
1
Each player rolls the 1–6 die once.
The player with the higher number
gets to choose what color he or she
wants to be and gets to take the first
turn. Then write your names and fill
in the color boxes at the top of your
record sheet.
2
Roll the die and use the number
you get to make one of the equations
in the grid true. There will be more
than one equation that will work for
any number. Write the number in the
box using your color.
3 Take turns until all the boxes are
filled. (If you roll a number you can’t
use, you lose that turn.) Both players
fill in every turn on their own record
sheets. Try to capture 3 or 4 boxes in a
row—across, up and down, or diagonally. After all the boxes are filled, help
each other use a calculator to check
the answers. Then circle the places
on the grid where you got 3 or 4 equations in a row and figure your scores.
4
Now play another round of the
game!
Number Corner Student Book
Jamal
NAME
DATE
Division Capture Game Sheet 1
COMPUTATIONAL FLUENCY
Jamal
Use a die
marked 1–6.
Scores:
Jamal 5
Tristan 3
Scoring:
3 in a row—1 point
4 in a row—2 points
Tristan
Round One
32 ÷
4
=8
30 ÷
5
=6
36 ÷
4
=9
21 ÷
3
=7
20 ÷
5
=4
12 ÷
1
= 12
40 ÷
5
=8
16 ÷
2
=8
24 ÷
2
= 12
27 ÷
3
=9
18 ÷
3
=6
24 ÷
4
=6
36 ÷
6
=6
8÷
2
=4
25 ÷
5 =5
4÷
1
=4
12 ÷
6
=2
6÷
1
=6
10 ÷
5
15 ÷
3
=5
=2
Here’s an example of a completed game.
Jamal would score 5 points because he got
2 sets of 4 in a row and 1 set of 3 in a row.
Tristen would get 3 points because she got
3 sets of 3 in a row.
Scoring
3 in a row
1 point
4 in a row
2 points
Make sure students see that there are three game sheets in their Student
Books, and that each sheet has room for them to play two rounds. The third
page features two challenge rounds that require a die marked 4–9 instead of
1–6. The first two sheets provide practice with division facts through 54 ÷ 6,
and the last sheet includes facts through 81 ÷ 9. Even fourth graders who are
quite fluent with their basic division facts will find the game strategies engaging, but you might encourage such students to start with the challenge version of the game on page 92 and possibly go on to design and use their own
game boards.
322 • Number Corner, Grade 4
April Overhead NC 8.4
Introducing Division Capture
1
Have each team choose a color and
fill in the boxes below to show what
they are. Then roll the 1–6 die to see
who goes first (high number starts).
2
Roll the die and use the number
you get to make one of the equations
below true. Write the number in the
box using your color.
3
Take turns until all the boxes are
filled. (If you roll a number you can’t
use, you lose that turn.) Try to capture 3 or 4 boxes in a row—across, up
and down, or diagonally. After all the
boxes are filled, circle the places on
the grid where you got 3 or 4 in a row,
and then add up both scores. You get 1
point for every set of 3 in a row and 2
points for every set of 4 in a row.
Students
Teacher
40 ÷
=8
21 ÷
=7
15 ÷
=3
7÷
=7
32 ÷
=8
10 ÷
= 10
20 ÷
=5
24 ÷
=8
35 ÷
=7
18 ÷
=3
14 ÷
=7
16 ÷
=4
36 ÷
=9
18 ÷
=9
25 ÷
=5
24 ÷
=4
30 ÷
=5
18 ÷
=6
36 ÷
=6
24 ÷
= 12
Scoring:
3 in a row—1 point
4 in a row—2 points
Number Corner
Students’ Score
Teacher’s Score
© The Math Learning Center
Number Corner Student Book
NAME
DATE
Division Capture Instructions
COMPUTATIONAL FLUENCY
1
Each player rolls the 1–6 die once.
The player with the higher number
gets to choose what color he or she
wants to be and gets to take the first
turn. Then write your names and fill
in the color boxes at the top of your
record sheet.
2
Roll the die and use the number
you get to make one of the equations
in the grid true. There will be more
than one equation that will work for
any number. Write the number in the
box using your color.
3
Take turns until all the boxes are
filled. (If you roll a number you can’t
use, you lose that turn.) Both players
fill in every turn on their own record
sheets. Try to capture 3 or 4 boxes in a
row—across, up and down, or diagonally. After all the boxes are filled, help
each other use a calculator to check
the answers. Then circle the places
on the grid where you got 3 or 4 equations in a row and figure your scores.
4
Now play another round of the
game!
Number Corner Student Book
Jamal
NAME
DATE
Division Capture Game Sheet 1
COMPUTATIONAL FLUENCY
Jamal
Round One
Use a die
marked 1–6.
Scores:
Jamal 5
Tristan 3
Scoring:
3 in a row—1 point
4 in a row—2 points
Tristan
32 ÷
4
=8
30 ÷
5
=6
36 ÷
4
=9
21 ÷
3
=7
20 ÷
5
=4
12 ÷
1
= 12
40 ÷
5
=8
16 ÷
2
=8
24 ÷
2
= 12
27 ÷
3
=9
18 ÷
3
=6
24 ÷
4
=6
36 ÷
6
=6
8÷
2
=4
25 ÷
5 =5
4÷
1
=4
12 ÷
6
=2
6÷
1
=6
10 ÷
5
15 ÷
3
=5
=2
Round Two
6÷
= 3 of
12 ÷a completed
= 4 36 ÷
= 9 30game.
÷
=5
Here’s
an example
Use a die
marked 1–6.
Jamal
÷
= 4 59 ÷points
=9
36 because
÷
= 12 18 ÷ he
= 3 got
Scores: would 16score
2 sets of 4 in15a÷ row
and =18 set
of= 35 in
a row.
= 3 24 ÷
10 ÷
30 ÷
=6
Tristen would get 3 points because she got
30 ÷
= 10 54 ÷
= 9 12 ÷
= 3 18 ÷
=9
3 sets of 3 in a row.
45 ÷
=9
1÷
=1
42 ÷
=7
21 ÷
=7
Scoring
3 in a row
1 point
4 in a row
2 points
© The Math Learning Center
Number Corner
85
Number Corner Student Book
NAME
DATE
Division Capture Game Sheet 1
COMPUTATIONAL FLUENCY
Scoring:
3 in a row—1 point
4 in a row—2 points
Round One
Use a die
marked 1–6.
Scores:
32 ÷
=8
30 ÷
=6
36 ÷
=9
21 ÷
=7
20 ÷
=4
12 ÷
= 12
40 ÷
=8
16 ÷
=8
24 ÷
= 12
27 ÷
=9
18 ÷
=6
24 ÷
=6
36 ÷
=6
8÷
=4
25 ÷
=5
4÷
=4
12 ÷
=2
6÷
=6
10 ÷
=2
15 ÷
=5
6÷
=3
12 ÷
36 ÷
=9
30 ÷
=5
16 ÷
=4
9÷
=9
36 ÷
= 12
18 ÷
=3
15 ÷
=3
24 ÷
=8
10 ÷
=5
30 ÷
=6
30 ÷
= 10
54 ÷
=9
12 ÷
=3
18 ÷
=9
45 ÷
=9
1÷
=1
42 ÷
=7
21 ÷
=7
Round Two
Use a die
marked 1–6.
Scores:
86
Number Corner
=4
© The Math Learning Center
Number Corner Student Book
NAME
DATE
Division Capture Game Sheet 2
COMPUTATIONAL FLUENCY
Scoring:
3 in a row—1 point
4 in a row—2 points
Round Three
Use a die
marked 1–6.
Scores:
7÷
=7
15 ÷
=5
24 ÷
=4
21 ÷
=7
20 ÷
=4
36 ÷
=6
27 ÷
=9
16 ÷
=8
28 ÷
=7
27 ÷
=9
14 ÷
=7
30 ÷
=6
20 ÷
= 10
8÷
=4
12 ÷
=3
4÷
=4
24 ÷
=8
6÷
=6
28 ÷
=7
45 ÷
=9
18 ÷
=9
24 ÷
=8
28 ÷
=7
35 ÷
=7
30 ÷
=5
9÷
=9
16 ÷
=8
40 ÷
=8
18 ÷
=3
27 ÷
=9
32 ÷
=8
14 ÷
=7
25 ÷
=5
36 ÷
=6
29 ÷
= 29
21 ÷
=7
64 ÷
= 64
42 ÷
=7
48 ÷
=8
Round Four
Use a die
marked 1–6.
Scores:
88
Number Corner
54 ÷
=9
© The Math Learning Center
Number Corner Student Book
NAME
DATE
Division Capture Game Sheet 3
COMPUTATIONAL FLUENCY
Scoring:
3 in a row—1 point
4 in a row—2 points
Round Five
Use a die
marked 4–9.
Scores:
36 ÷
=6
14 ÷
=2
36 ÷
=4
28 ÷
=4
63 ÷
=7
27 ÷
=3
21 ÷
=3
18 ÷
=2
20 ÷
=4
24 ÷
=6
30 ÷
=6
36 ÷
=6
72 ÷
=8
32 ÷
=4
54 ÷
=6
48 ÷
=6
45 ÷
=9
40 ÷
=5
36 ÷
=9
42 ÷
=7
45 ÷
=9
42 ÷
=7
72 ÷
=8
81 ÷
=9
48 ÷
=6
28 ÷
=4
50 ÷
= 10
36 ÷
=9
24 ÷
=6
63 ÷
=7
48 ÷
=8
35 ÷
=5
40 ÷
=5
40 ÷
=8
40 ÷
= 10
56 ÷
=7
54 ÷
=9
32 ÷
=4
54 ÷
=6
42 ÷
=6
Round Six
Use a die
marked 4–9.
Scores:
92
Number Corner
© The Math Learning Center
C
M
Y
CM
MY
CY
CMY
K
Starting
Position
1
C
M
Y
CM
MY
CY
CMY
K
2
C
M
Y
CM
MY
CY
CMY
K
3
C
M
Y
CM
MY
CY
CMY
K
4
C
M
Y
CM
MY
CY
CMY
K
5
C
M
Y
CM
MY
CY
CMY
K
6
7
C
M
Y
CM
MY
CY
CMY
K
8
C
M
Y
CM
MY
CY
CMY
K
C
M
Y
CM
MY
CY
CMY
K
9
C
M
Y
CM
MY
CY
CMY
K
10
C
M
Y
CM
MY
CY
CMY
K
11
C
M
Y
CM
MY
CY
CMY
K
12
C
M
Y
CM
MY
CY
CMY
K
13
C
M
Y
CM
MY
CY
CMY
K
14
15
C
M
Y
CM
MY
CY
CMY
K
C
M
Y
CM
MY
CY
CMY
K
16
C
M
Y
CM
MY
CY
CMY
K
17
C
M
Y
CM
MY
CY
CMY
K
18
C
M
Y
CM
MY
CY
CMY
K
19
C
M
Y
CM
MY
CY
CMY
K
20
C
M
Y
CM
MY
CY
CMY
K
21
C
M
Y
CM
MY
CY
CMY
K
22
23
C
M
Y
CM
MY
CY
CMY
K
24
C
M
Y
CM
MY
CY
CMY
K
C
M
Y
CM
MY
CY
CMY
K
25
C
M
Y
CM
MY
CY
CMY
K
26
C
M
Y
CM
MY
CY
CMY
K
27
C
M
Y
CM
MY
CY
CMY
K
28
C
M
Y
CM
MY
CY
CMY
K
29
C
M
Y
CM
MY
CY
CMY
K
30
C
M
Y
CM
MY
CY
CMY
K
31
Number Corner, Grade 4 • 107
6 inches
6 inches
6 inches
6 inches
×6
12
×6
×6
×6
×6
15
16
17
18
1 ´
2
5´
6
11´
126˝
132˝
×6
21
22
Calendar Collector
10´
10
120˝
×6
20
21
=
2
114˝
×6
9´
19
=9
2
1´
2
1 ´
2
´
72´
1
1´
2
8´
17
1
=8
2
2
15
2 =
7´
13
=
2
6´
11
=
2
1 ´
2
5 21 ´
4´
4
3´
1 ´
=3
2
2´
1 ´
=2
2
9
=
2
7
2
5
2
3
2
1´
=1
1 ´
2
Feet
108˝
102˝
96˝
90˝
84˝
78˝
72˝
66˝
60˝
54˝
48˝
36˝
42˝
24˝
30˝
18˝
12˝
6˝
Inches
×6
19
×6
14
×6
×6
13
×6
11
×6
×6
×6
×6
×6
10
9
7
8
6
×6
×6
3
4
5
×6
×6
×6
Inches, Feet & Yards Record Sheet
6 inches
6 inches
2
1
Day
6 inches
6 inches
Yards
3 yards
2 yards
1 yard
1 foot
1 foot
1 foot
1 foot
6 inches
6 inches
6 inches
6 inches
1 ya rd
1 ya rd
1 ya rd
1 ya rd
21
14
7
Sunday
6 inches
6 inches
6 inches
6 inches
1
22
15
8
Starting
Position
Monday
2 feet
2 feet
2 feet
2 feet
6 inches
6 inches
6 inches
3 feet
3 feet
3 feet
3 feet
16
9
2
17
10
3
Wednesday
18
11
4
Thursday
November
Tuesday
6 inches
6 inches
6 inches
5
19
12
Friday
20
13
6
Saturday
1st
Starting Position
Move (How did the triangle get there?)
November Calendar Record Sheet
Quadrant
Calendar Grid
1
Date
November
November Calendar Grid
CALENDAR GRID
The Tumbling Triangle
Overview
You’ll need
This month a single triangle appears in a
different location on a quadrant grid each
day, going through a predictable series of
transformations every 4 days: slide, slide,
slide turn, flip. This pattern provides ongoing opportunities to develop the language
and concepts of motion geometry.
H Quadrant Grid & Triangles (Overhead
NC 3.1, cut out triangles ahead of time)
H Filled Calendar Grid (Overhead NC 3.2,
optional)
H Quadrant Grids (Blackline NC 3.1, halfclass set, cut in half)
H Triangles (Blackline NC 3.2, one-third
class set on colored paper, cut apart)
Frequency
Update the Calendar Grid each day and
share observations and predictions twice
a week.
Skills & Concepts
H November Calendar Grid Record
Sheet, pages 1 and 2 (Blacklines
NC 3.5 and 3.6, 1 copy each taped
together, see Advance Preparation)
H describing, extending, and making
verbal and written generalizations
about geometric patterns to make predictions and solve problems
H Shapes in Motion, pages 1 and 2
(Number Corner Student Book, pages
31 and 32)
H identifying, describing, comparing,
and classifying triangles by attributes
of their sides and angles
H Day, Month, and Year markers
H predicting and describing the results
of performing flips, slides, and turns
H identifying right, acute, and obtuse
angles in geometric figures
Starting
Position
1
2
H Calendar Grid pocket chart
H Tumbling Triangle calendar markers
H scissors
H black crayons or markers
Advance Preparation Before you conduct your first Calendar Grid Workout this
month, run a half-class set of the Quadrant
Grids and a one-third class set of Triangles.
Cut sheets in halves and in thirds respectively so that each student gets 1 quadrant
grid and a set of 4 paper triangles. Also run
1 copy each of Blacklines NC 3.5 and 3.6.
Trim and attach the two sheets to form one
long record sheet. Post the record sheet
before your second Calendar Grid Workout
this month. You can run a class set of Blacklines NC 3.5 and 3.6 if you want students to
fill in the record sheet independently.
Number Corner, Grade 4 • 115
November
Calendar Grid The Tumbling Triangle (cont.)
Motion Geometry: Background Information for the Teacher In motion geometry, more formally called transformational geometry, there are four different
ways to move a figure from one place to another without changing its size or shape.
This month’s Calendar Grid features 3 of the 4 motions: translations, rotations, and
reflections (informally called slides, turns, and flips). Mathematical definitions of
the terms are provided below for your reference. We don’t expect fourth graders to
master the ideas or vocabulary in any formal way, but you will be better able to
facilitate their explorations and discoveries if you use the vocabulary accurately and
consistently. We recommend that you allow students to describe the calendar markers using the language that makes the best sense to them, but that you use formal
terminology in reflecting their ideas back to them and encourage them to use the
words translation, rotation, reflection, and congruent by the end of the month.
MOTION GEOMETRY
Translation or Slide
A translation is a movement of a point or a figure
a certain distance in a certain direction. In the
example at the right, figure D is a translation image*
of figure C. The direction is due east (or right or
horizontal) and the distance is AB.
Rotation or Turn
A rotation is a movement of a point or a figure a
certain angle measure about a central point (the
center). In the example at the right, figure D is a
rotation image of figure C. It is a 90° clockwise rotation about center O. (The measure of angle AOB is
90 degrees and AO = OB.)
Reflection or Flip
When you reflect a point or a figure across a line
(called a line of reflection), the new figure looks like
a reflection or mirror image of the original. In the
example at the right, point B is the reflection image
of point A over line m because
1. AB is perpendicular to m
2. and the distance from B to m equals the distance
from A to m.
Figure D is the reflection image of figure C.
Congruent Figures
Figures that have exactly the same shape and same
size are congruent. Two figures are congruent if
there is a combination of reflections, rotations, and
translations to map one onto the other. If figures are
congruent, all their corresponding parts (angles,
sides) are congruent, as in figures C and D at right
and in all examples above.
D
C
B
A
C
B
D
A
O
m
A
C
B
D
C
D
* The image of a figure is the set of all the points of the figure for a given translation, rotation, or reflection.
116 • Number Corner, Grade 4
November
Calendar Grid The Tumbling Triangle (cont.)
Week 1 Introducing the Calendar Grid
Invite a volunteer to post all the markers for the days that have passed so far
this month, including today, and ask students to examine the marker(s) quietly for a minute or two. Then invite volunteers to describe what they see.
Starting
Position
1
Students It’s a triangle this month. And it says starting position. I
wonder what that means.
Maybe it’s going to move around or something.
It’s a blue triangle, kind of turquoise, and it has a dot in the corner.
It’s on a grid.
There are 2 black lines that have arrows on both ends and cross in the
middle.
The triangle fits right in that corner.
After students have had a chance to offer some initial observations, explain
that you want to offer them some hands-on experience with the markers
this month. Then give each student a copy of the grid and 4 triangles. Before
they cut out their triangles, ask them to take a moment to study the triangles.
What else can they say about these triangles that hasn’t already been said?
Have them brainstorm in pairs and then invite students to share their observations with the class.
November Blackline NC 3.2 Run a one-third class set on colored paper. Cut apart so each student gets 4 triangles.
Triangles
Susie There are 4 of them and they’re all the same size and shape.
Keith They’re not the same kind of triangle as the pattern block.
Sage Their sides aren’t equal. None of them look the same length.
Connor They have a right angle.
Teacher Connor has mentioned that these triangles have right angles,
and Sage has suggested that their sides are different lengths. Susie has
Number Corner, Grade 4 • 117
November
Calendar Grid The Tumbling Triangle (cont.)
suggested that they’re all congruent, the same size and the same shape.
Take a minute to talk with the people next to you about these observations. Do you agree? You can cut one of them out if that will help you
test these ideas.
If the issue of symmetry doesn’t come up, invite students to consider
whether or not this triangle is symmetrical. They might cut one out and try
folding it to find lines of symmetry. As they work, they will discover that this
right triangle has no symmetry, because there is no way to fold it so that the
two parts match exactly.
After a bit of discussion, ask students to cut out all 4 triangles and use a black
crayon or marker to make a dot on the back of each, in the very same place
as the one on the front, so that the triangles look just the same on the front
and back. Then give your students the last few minutes to experiment with
their triangles on the grid. What kinds of designs can they make by moving the triangles around and positioning them on the grid in various ways?
No further discussion is necessary, although many students will be eager to
share their favorite designs with classmates. This hands-on experience at the
beginning of the month will help them make observations and predictions
about the calendar markers in the days to come.
At the end of the workout, have students either store their grids and triangles
in a safe place or collect them; they’ll be using them again during Calendar
Grid Workouts this month.
Introducing the Calendar Grid Record Sheet
When you return to the Calendar Grid several days later, there will be
enough evidence to provide students with opportunities to discuss the relationships between the markers that have been posted so far, and to make predictions about what might come next.
November
Sunday
Monday
Starting
Position
1
Tuesday
2
Wednesday
3
Thursday
4
Saturday
Friday
5
Note This month’s calendar markers may be difficult for some students to see at a
distance. If this is the case in your classroom, you might want to display the Filled
Calendar overhead and mask all but the markers through the current date.
118 • Number Corner, Grade 4
November
Calendar Grid The Tumbling Triangle (cont.)
After students have made some initial observations about the markers, have
them get out the grid and one of the triangles they cut out during the first
workout. Ask them to take a few minutes to move the triangle on their own
grid from one position to the next. What do they have to do to get it there?
What words can they use to describe each day’s new location? Tell students
that mathematicians refer to the 4 quadrants by number, as shown in the figure below. You may want to sketch this diagram on the whiteboard. Encourage students to use the quadrant numbers, as well as words like up, down,
right, left, above, and below to describe the triangle’s position and movements.
November Blackline NC 3.1 Run a half-class set and cut in half.
Quadrant Grids
2
1
3
4
four quadrants
After they’ve had a few minutes to work on their own, have students share
their discoveries in partners, and then invite a few up to the overhead to
share with the whole group, using transparencies of the grid and triangle. It’s
quite likely that students will use a combination of movements to get the triangle from one place to another, which is perfectly fine.
Starting
Position
1
2
Cheyenne To get it from where it is on marker to 1 to where it is on
marker 2, I moved it up 1 and over 1, like this.
November Overhead NC 3.1
November Overhead NC 3.1
Quadrant Grid & Triangles
Quadrant Grid & Triangles
up one
over one
Number Corner, Grade 4 • 119
November
Calendar Grid The Tumbling Triangle (cont.)
Antoine I did it differently. I went over 1 and then up 1.
Teacher Was anyone able to get it there in just one motion? No? …
Take a few moments to see if you can do it in one motion….
Lilly Do you have to just go on the grid lines? You could just go diagonally, like this.
November Overhead NC 3.1
Quadrant Grid & Triangles
Cut out these triangles so students can use them to show motions and
predictions on the grid above.
Teacher All of the ways you described use slides, and Lilly showed
us how we can use just one slide, so we’ll describe this movement as a
slide. Another word for slide is translate, so I’ll write that here too.
As students share, fill in the Calendar Grid Record Sheet up through today’s
date with the day, the quadrant, and a description of the triangle’s movement.
November Calendar Record Sheet
Date
Quadrant
Move (How did the triangle get there?)
1
1st
Starting Position
2
1st
Slide (Translate)
3
2nd
Slide (Translate)
4
2nd
Slide, Turn (Translate, Rotate)
5
3rd
Flip (Reflect)
Note To get the triangle from its location on marker 3 to its location on marker
4, most students will rotate the triangle using the black dot as a center of rotation,
and then slide it into place (or vice versa). If so, record the motion as shown above.
It is possible to reposition the triangle using a single 90-degree rotation about point
A, shown below. If you cut out a triangle, place it on the grid, put your pencil point
on point A, and then rotate, you can see this for yourself. It is unlikely that fourth
graders will identify any of the points of rotation (some of which are inside the trianglue or tape page 2 here
120 • Number Corner, Grade 4
November
Calendar Grid The Tumbling Triangle (cont.)
gles and some of which are not) that will allow them to reposition the triangles with a
single rotation, but you might extend the challenge to some if it seems appropriate.
A
4
3
End the discussion by asking children to make predictions about the next
day’s marker. Have students use their triangles and grids to show their predictions, and have a couple of students share their thinking and explain their
reasoning at the overhead.
Weeks 2 & 3 Discussing the Pattern & Motions
Ask a volunteer to update the Calendar Grid and Record Sheet daily, and discuss it as a class at least once a week, twice a week if possible. You may want
to leave a small sketch of the quadrant numbers posted beside the record
sheet. Continue to have students use their own paper grid and triangles to
make observations and predictions, and have them use the overhead grid and
triangles when sharing with the whole class.
You’ll probably need to let students know that when they’re flipping the triangle from one quadrant to another on these markers, the heavy quadrant lines
serve as the lines of reflection. For example, to get from marker 8 to marker
9, a student might initially flip the triangle over its short side and then slide it
up as shown below.
8
9
start
flip over short side
slide up
marker 9 position
end
Let them know, however, that when they flip or reflect a shape, they can
flip or reflect it over any line. The line of reflection need not be a side of the
shape itself, and in this pattern, the line of reflection, about which they can
perform a single flip, is always one of the quadrant grid lines.
Number Corner, Grade 4 • 121
November
Calendar Grid The Tumbling Triangle (cont.)
line of reflection
You can use the following questions throughout the month to get at the big
ideas. In the end, the goal isn’t so much for students to “get” the pattern, but
to begin to develop deep understandings about shapes and how they can be
moved from place to place.
• What do you notice about the markers, by themselves and as a collection?
• Use your own grid and triangle to copy the position of the triangle on
today’s marker. Describe the location of the triangle as thoroughly as you
can. Which quadrant is it in now? Where, exactly, is it in the quadrant?
Where is it in relation to the heavy black lines? How did you have to move
it from yesterday’s position to get it to where it is today?
• Use your own grid and triangle to make a prediction about the location of
tomorrow’s marker and explain your thinking. What is it about the pattern
so far that would support your prediction?
Week 4 Completing Pages 31 & 32 Independently
Toward the end of the month, have students complete pages 31 and 32,
Shapes in Motion, or assign it for homework, as it will consolidate some of the
explorations they’ve done on the Calendar Grid.
Number Corner Student Book
Number Corner Student Book
NAME
DATE
Shapes in Motion page 1 of 2
NAME
DATE
Shapes in Motion page 2 of 2
CALENDAR GRID
CALENDAR GRID
1
How would you move the triangle to get it from its position on the first grid to
its new position on the second grid?
a
b
c
d
3 Which pair of figures above does not show a translation (slide)? Fill in the
bubble to show your answer.
�
first grid
2a
second grid
Draw the next 2 shapes in the pattern on the grid.
b Explain how you have to move the shape each time to make the pattern work.
122 • Number Corner, Grade 4
a
�
b
�
c
�
d
4 On the grid below, draw a reflection of this triangle.
Number Corner Student Book
NAME
DATE
Shapes in Motion page 1 of 2
CALENDAR GRID
1
How would you move the triangle to get it from its position on the first grid to
its new position on the second grid?
first grid
2a
second grid
Draw the next 2 shapes in the pattern on the grid.
b Explain how you have to move the shape each time to make the pattern work.
© The Math Learning Center
Number Corner
31
Number Corner Student Book
NAME
DATE
Shapes in Motion page 2 of 2
CALENDAR GRID
a
b
c
d
3 Which pair of figures above does not show a translation (slide)? Fill in the
bubble to show your answer.

a

b

c

d
4 On the grid below, draw a reflection of this triangle.
32
Number Corner
© The Math Learning Center
November Blackline NC 3.1 Run a half-class set and cut in half.
Quadrant Grids
© The Math Learning Center
Number Corner
November Blackline NC 3.2 Run a one-third class set on colored paper. Cut apart so each student gets 4 triangles.
Triangles
Number Corner
© The Math Learning Center
November Blackline NC 3.5 Run 1 copy, trim, and attach Blackline NC 3.6 to create one long record sheet.
November Calendar Grid Record Sheet page 1 of 2
November Calendar Record Sheet
Date
Quadrant
Move (How did the triangle get there?)
1
1st
Starting Position
Glue or tape page 2 here.
© The Math Learning Center
Number Corner
November Blackline NC 3.6 Run 1 copy, trim, and attach to Blackline NC 3.5 to create one long record sheet.
November Calendar Grid Record Sheet page 2 of 2
Number Corner
© The Math Learning Center