Build Towers to Show Doubles and Near Doubles.
Display several of the Doubles Cards you prepared.
-
Which sums will be odd and which will be even?
-
Can you tell without solving the problem?
Direct students to the Doubles and Near Doubles
page from the Student Activity Book. Display the
Doubles and Near Doubles chart prepared before the
class. Choose one of the
Doubles Cards
to display
and write the number sentence in the first space in
the "Doubles Number Sentence" column. Ask students
to also write this number sentence in the first
space on their page and then work with a partner to
build doubles towers of connecting cubes to illustrate
this number sentence and find the sum. Each
student should have his or her own set of towers.
For this example, the number sentence 8 + 8 = _____ will be used.
-
What do you notice about the doubles towers that
you built? (Possible response: Both towers have
the same number of cubes in them. Each tower
has 8 cubes in it.)
-
Using your doubles towers, tell your partner what
strategies you can use to find the sum of 8 + 8.
(Possible responses: There are eight cubes in
each tower so I started on eight and then counted
on using the cubes in the second tower. I got 16.
Or, I put the two towers next to each other and
then counted by twos to 16. Or, I took two cubes
from one tower and added them to the other
tower to make ten. Then I added ten and six and
got 16.)
Encourage several students to share their strategies
with the class and then add the sum to the number
sentence on the chart before asking:
-
Is the sum 16 even or odd? (even) Tell your partner
how you know. (Possible responses: When I
put the two towers next to each other, each cube
has a partner. Or, when you put the two towers
next to each other they are the same size. Or,
when you count by twos you say 16, so it is
even.)
After several students share their reasoning, write
the word "even" in the second column of the chart
and ask students to do the same. Now direct one person
in each pair of students to add one cube to only
one of his or her towers.
-
If you add one cube to only one of your towers of
8, how many cubes do you have now? (17)
Explain the strategy you used to find your answer
to your partner. (Possible responses: I knew there
were already 16 cubes so I just counted one more
to 17. Or, I know 8 + 8 = 16, so 8 + 9 = 17.)
-
These towers are called doubles-plus-one towers.
Compare your doubles-plus-one towers with your
partners' doubles towers. What do you notice?
(Possible response: The doubles towers are the
same size but in the doubles-plus-one towers, one
tower is bigger than the other.)
-
What number sentence can you write on your
chart that shows the sum of the doubles-plus-one
towers? (Possible responses: 8 + 9 = 17;
9 + 8 = 17; or
8 + 8 + 1 = 17)
-
Is 17 even or odd? (odd) How do you know?
(Possible responses: There is one cube that does
not have a partner. Or, when you put the towers
next to each other and count by twos, you will
have one cube left over.)
Remind students to fill in their charts with a doubles-plus-one number sentence and the word “odd”
for the sum as you add the information to the display
chart.
Ask all of the students to show their doubles towers
for 8 + 8 = 16 again. This time ask one person in
each pair to subtract one cube from one of his or her
two towers.
-
If you take one cube from one of your towers of 8,
how many cubes do you have now? (15) Explain
the strategy you used to decide to your partner.
(Possible responses: I know that 8 + 8 = 16, so
when I took one cube away I counted back one
from 16 and got 15. Or, since I know 8 + 8 = 16,
I know 8 + 7 will be one less, or 15.)
-
These are doubles-minus-one towers. Compare
your doubles-minus-one towers with your partners’
doubles towers. What do you notice?
(Possible responses: The doubles towers are the
same size but one tower is shorter than the other
in the doubles-minus-one towers.)
-
What number sentences can you write on your
chart to show the sum of the doubles-minus-one
towers? (Possible responses: 8 + 7 = 15;
7 + 8 = 15; or 8 + 8 − 1 = 15)
-
Is 15 even or odd? (odd) Explain how you decided
to your partner. (Possible response: When I put
the towers next to each other there was one cube
without a partner. Or, when I counted the cubes
by twos I counted to 14 and then there was one
cube left.)
Ask students to fill in the number sentence for doubles-
minus-one and the word "odd" for the sum on
their charts, as you fill in the chart display.
Tell students that they are going to work with their
partner to fill in the other rows on their charts.
Distribute the remaining Doubles Cards so that each
student has two different cards. Students will work
with their partners to complete a row for each of the
four cards. As students are working, circulate
through the classroom asking several volunteers to
use their charts to help you fill in the next three or
four rows on the displayed chart. See Figure 3.
Looking for Patterns. After students have had time
to work on their individual charts with their partners,
bring their attention back to the display of the class
chart. Ask students to look closely at the chart and
identify any patterns they can see. See the Sample
Dialog.
Use this dialog to help guide a discussion about the patterns
on the Doubles and Near Doubles chart.
Teacher: What patterns do you notice in the table?
Jessie: When you add doubles the answer is always even.
But it is always an odd answer when you add doubles
+1 or doubles -1.
Teacher: Jessie, why do you think that the answer is
always even when you are adding doubles?
Jessie: When you are adding doubles and you make towers
with cubes each cube will always have a partner.
Teacher: Does anyone see any other patterns? What do you
notice about the sums in the Doubles and Doubles +1
columns?
Frank: The answers in the Doubles +1 column are always
one bigger than the answers in the Doubles column.
Teacher: What about the sums in the Doubles −1 and
Doubles columns?
Frank: In the Doubles −1 column the sums are always one
less than in the Doubles column.
Teacher: Do you see any pattern in the numbers you are
adding in the Doubles column?
Jerome: When you are adding doubles both of the numbers
in the problem are either odd or even.
Teacher: Jerome, can you explain what you mean by that?
Jerome: When you are adding doubles the two numbers are
the same so they are both even or they are both odd. If
you add 4 + 4 both of the numbers are 4 and that
means they are both even numbers, but if you add
5 + 5 the numbers are both odd.
Teacher: Good. What do you notice about the numbers you
are adding in the doubles 1 and doubles −1
columns?
Linda: In those problems one of the numbers is always even
and the other is always odd. The answer is also always
an odd number.
Teacher: We have a lot of information about doubles and
near doubles from our chart. Let's use that
information to make some predictions about the next
row. Ana, what is on your card?
Ana: 6 + 6
Teacher: [Writes 6 + 6 in the Doubles column.] Before Ana
tells us how to fill in the rest of this row, let's make
some predictions.
Ask the class to predict what will go in each column
of the table and then have the student whose card
was chosen check the results with his or her tower.
Continue this process, filling in several more rows
on the table and then ask:
-
Suppose I wanted to double 20, would my answer
be odd or even? (even) Explain how you decided.
(Possible response: When you double a number
you can build two towers and they are both the
same size so each cube will have a partner.)
-
Since 20 + 20 is equal to 40, how can you use that
to find the answer for 20 + 21? (Possible
response:
20 + 21 is a double +1 so the answer
will be one more than 40 or 41.) Is the answer
odd or even? (odd) How do you know? (Possible
response: When you have doubles you can count
by twos and there are no leftovers but if you have
doubles +1 and count by twos there will be one
left.)
-
How can you use 20 + 20 to help you find the
answer for 20 + 19? (Possible response: The
answer will be 39, because 20 + 20 = 40, so
20 + 19 will be one less or 39.)
Have students complete Check-In: Questions 1–3
on the Doubles and Near Doubles pages in the
Student Activity Book. Before students begin working,
review Math Practice 1, Know the problem, and
Math Practice 5 Show my work, using the Math
Practices page in the Student Activity Book
Reference section.
-
What do you need to remember when you are
reading each problem? (Possible response: You
have to read carefully to know what numbers you
will use and what the question is asking you to
do. You need to follow the directions carefully.)
-
What can you do to help others understand your
thinking? (Possible response: You can write a
number sentence, draw a picture, and use words
to tell how you found your answer.)
Use
Check-In: Questions 1–3 with Feedback Box on the
Doubles and Near Doubles pages in the
Student Activity Book
to assess students' abilities to represent doubles and near
doubles using counters, pictures, and number sentences
[E1]; use reasoning strategies (e.g., using doubles) to solve
problems with sums between 10 and 20 [E5]; know the
problem [MPE1]; and show work [MPE5].
The
Doubles, Doubles +1, Doubles −1 game can be placed in
a center and can be used to provide additional targeted
practice with doubles, doubles +1, and doubles −1.
