NS2-45
Skip Counting
Pages 1-8
Goals
Students will skip count by 2s, 5s, or 10s from 0 to 100, and back from 100 to 0.
Students will skip count by 5s starting at multiples of 5, and by 2s or 10s starting at any
number.
PRIOR KNOWLEDGE REQUIRED
can count by 1s
can use a hundreds chart to count
can use a hundreds chart to add or subtract 10
can use a number line
can measure centimetres with a ruler
VOCABULARY
skip
skip counting
count back
how many
hundreds chart
number line
more/ less
ones digit
tens digit
centimetres
MATERIALS
flip chart
rulers
metre sticks
large hundreds chart in school yard
cards numbered 1 through to the number of students
counters
BLM A Larger Hundreds Chart (p xxx)
CURRICULUM EXPECTATIONS
Ontario: 1m21, 2m4, 2m6, 2m7, 2m19, 2m20,
WNCP: 1N1, 1N3, 2N1, [CN, C, V, R]
Introduce skip counting by skipping every second number. SAY: I want to count
every second number. This is called skip counting by 2s. Write the words “skip counting”
on the board. To demonstrate, SAY: 0 (loudly), skip 1 (quietly), 2 (loudly), skip 3
(quietly), 4 (loudly), and so on to 10. Repeat with students. Explain that when you skip
count by 2 you add 2 instead of 1 to find the next number.
Skip count using a chart. Draw the first row of a hundreds chart on a flip chart. Have
the class slowly skip count by 2s. Colour the numbers as the class says them:
1
2
3
4
5
6
7
8
9
10
Cover up the chart and ask students if 3 is a number they say when counting by 2.
(thumbs up for yes, thumbs down for no) Then uncover the chart to check. Repeat for
other random numbers. Eventually stop checking answers and increase the speed at
which you say the numbers.
Count by 2 up to 100. Draw the first two rows of a hundreds chart on the board and ask
a volunteer to show skip counting by 2 by shading every second square. Ask students if
they see anything the same about the numbers in the first row and the numbers in the
second row. PSS Looking for a pattern. ASK: If I can skip count by 2 up to 10, does that
help me skip count up to 20? How? (The numbers up to 20 have the same ones digit as
the numbers up to 10.) Does it help me skip count to 100? (yes, the ones digits are
again the same as skip counting up to 10)
Activity 1. Skip count on a large hundreds chart in the school yard. Have students
hop on every second number, starting at 2 and chant the numbers as they land on them.
When the first student says “12,” the next student says “2.”
Count back by 2s from 100 to 0. Teach students to skip count back by 2s from 10 to 0
by memorizing the sequence. Then teach them to skip count back by 2s from 20 to 10
by using the same pattern in the ones digits. Repeat for 30 to 20, and so on. Repeat
Activity 1 but start at 100 and go backwards.
Count by 2s from 1. As a class, count by 2s starting at 1. SAY: 1 (loudly), skip 2
(quietly), 3 (loudly), and so on. Then have students count by 2s starting at 1 on the first 2
rows of a hundreds chart.
Activity 2. Catch. (See NS Part 1 – Introduction) “Throw” a number from 0 to 100,
(include both odd and even numbers), to one student after another. The student says the
next number counting by 2s (i.e. the number that is two more than).
Count back by 2s starting from any number. Repeat Activity 1 but start at 99 and go
backwards. Then teach this the same way you taught counting back by 2s from 100.
Finally, repeat Activity 2, but this time the student says the next number counting back
by 2s (i.e. the number that is two less than).
Count by 10s from 0 to 100 using a hundreds chart. Remind students that to add 10,
they simply move down a row. Emphasize that counting by 10s is the same as counting
by 1s, except that they just add a 0 to the numbers they say when counting by 1.
1
10
2
20
3
30
4
40
5
50
6
60
7
70
8
80
9
90
10
100
It may be helpful for some students to notice the similarities in the sound of the numbers
as well: e.g., three and thirty, four and forty. Repeat the previous activities counting by
10s instead of by 2s. Then have students check their answers for what comes next by
providing them with the number that comes after that. PSS Reflecting on the
reasonableness of an answer
EXAMPLES: 30 ____ 50; 10 _____ 30; 70 _____90; 60 ______80. Students should
guess that 40 comes after 30 and then check that 50 comes after 40 when counting by
10s.
Provide students with BLM A Larger Hundreds Chart and have them count by 10s from
various numbers (all with ones digit 0). Remind students that they can move down a row
to add 10. EXAMPLE: Start at 30, and count 30, 40, 50, 60, 70, 80, 90, 100.
Count back by 10s from 100 to 0. Using a hundreds chart, move up a row instead of
down a row. Or count back by 1s instead of forward by 1s to help.
Count by 10s starting from any number. When counting forwards or backwards, the
last digit stays the same. The number of tens goes up or down by 1. Start at any number
and go up or down a row in a hundreds chart.
Use a metre stick to count forward or backward by 10s. Use a ruler at least 10 cm
long, preferably exactly 10 cm long. EXAMPLE: To count back by 10s from 83, place the
10 cm mark on the 83. Where is the 0 mark? (on 73) Continue counting back in this way.
(63, 53, 43, 33, 23, 13, 3)
Compare using a hundreds chart and a metre stick. ASK: Which way makes
counting back by 10s easier? Why? PSS – Selecting tools and strategies
Count by 5s from 0 to 100. To count by 5s, repeat the lesson for counting by 10s.
Emphasize that counting by 5s is easy once they know how to count by 10s. SAY: After
you count 0, 5, 10, 15, say the same numbers you would say counting by 10s, and then
repeat that number once more with a “five” after it: 20, 25, 30, 35, and so on.
Activity 3. PSS – Organizing data, Visualization
Give each student one number card from 1 to the number of students. Have students
order themselves starting at the student with card number 1. Then tell students to
“shuffle” themselves. When students are well-shuffled, have them order themselves by
first placing those who have cards that count by 5s (0, 5, 10, 15, 20); the remaining
students can then place themselves in-between where they belong. Discuss which way
was easier, which way took longer, and why.
Count back by 5s from 20 to 0. Teach students to memorize the sequence (20, 15, 10,
5, 0). As a class, say the sequence forwards to 10 (0, 5, 10) then backwards (10, 5, 0),
then forwards to 15 (0, 5, 10, 15), then backwards (15, 10, 5, 0), then forwards to 20 (0,
5, 10, 15, 20), then backwards (20, 15, 10, 5, 0).
Count back by 5s from 100 to 0. Emphasize that counting back by 5s is easy once
they know how to count back by 10s; say the same numbers you would say counting
back by 10s, but before saying a number, say it first with a “five” after it. For example,
after saying 30, 20 would be next counting by 10s, so say 25 first, then 20. Show this on
a number line or hundreds chart.
Grouping objects to count them. Show students a large pile of counters, say 57
counters, and tell them that you want to count how many there are. Start by counting 1,
2, 3, 4, and so on. Demonstrate making a mistake partway through and explain that you
need to start over because you forgot where you are in counting. Then count 5 at a time
and put them in groups of 5. Invite volunteers to help. Explain that you find it easier to
count to 5 because at least you won’t get lost in the counting. Once you’ve grouped the
pennies into groups of 5, with 2 leftover, explain that now you can use skip counting to
find how many there are. Since there are 5 in each group, you can skip count by 5s. Do
this together as a class. 5, 10, 15, 20, and so on, until 55. SAY: There are 55 here and 2
more. How many is that? Now we have to count by 1s because we no longer have
groups of 5. Write the counting sequence on the board: 5, 10, 15, 20, 25, 30, 35, 40, 45,
50, 55, 56, 57.
Extensions:
1. Find the mistake in skip counting. Ontario teachers might use the optional lesson
PA2-14: Finding Mistakes as an extension here. Do not use it as an extension here,
though, if you plan to teach it later.
2. Combine skip counting by 2s and skip counting by 10s to skip count by 20s. To
count by 20s, count by 10, but skip every second 10. (SAY: skip 10 (quietly), say 20
(loudly), skip 30 (quietly), say 40 (loudly), and so on.)
Bonus: Have students skip count from 400 to 500 and then from 560 to 660.
3. On BLM Skip Counting (p xxx), students will discover that the numbers they say
when counting by 10s are the numbers they say both when counting by 2s and by 5s.
4. Using skip counting to add many numbers. Have students pair up the numbers
that add to 5 or 10 to add many numbers.
EXAMPLES: 4 + 1 + 3 + 2 + 5 + 2 + 3 = 5 + 5 + 5 + 5 = 20 (skip count 5, 10, 15, 20)
8 + 2 + 3 + 7 + 6 + 4 + 10 + 10 + 5 + 5 + 1 + 9 = 10 + 10 + 10 + 10 + 10 + 10 + 10 = 70
(skip count 10, 20, 30, 40, 50, 60, 70)
Provide BLM Adding Many Numbers (p xxx).
5. PSS – Guessing, checking and revising Teach students what to count by given the
first and last numbers, and the number of spaces in between. Draw a number line like
this:
10
14
SAY: I want to skip count from 10 to 14 and I want to say only one number in-between. If
I count by 1s, I know that 11 is right after 10, but 14 doesn’t come right after 11, so that
won’t work. ASK: What should I skip count by if I want the same number to come right
after 10 and right before 14? (count by 2s) Check this answer by skip counting from 10
to 14. Repeat with more examples, gradually leaving more spaces between numbers.
Have students progress as follows: first choose between skip counting by 2s or 5s, then
choose between skip counting by 5s or 10s; finally choose between skip counting by 2s,
5s or 10s. Students should guess what to skip count by and check their guess – if, when
skip counting by their guess, they don’t get to the end number, students should decide
whether to skip count by a higher or lower number based on the results.
AT HOME House numbers
CONNECTION—Literature,
Two of Everything by L.T. Hong A Chinese folktale counts everything by 2s.
NS2-46
Closer To
Pages 9-12
Goals
Students will determine the closest ten by using the distance on a number line.
PRIOR KNOWLEDGE REQUIRED
understands the concept of distance
can count to 100
knows how to use number lines
VOCABULARY
close/ closer/ closest
more/ less
between
far apart
further
equally
MATERIALS
large cards numbered 0 through 10
a large visible hundreds chart (e.g. a pocket chart)
BLM Closest (p xxx)
BLM Number Lines 0 to 10 (p xxx)
BLM Closer to 0 or 10 (p xxx)
BLM Closer to 40 or 50 (p xxx)
CURRICULUM EXPECTATIONS
Ontario: 2m1, 2m6, 2m7, 2m14
WNCP: 2N6, [V, C, R]
Review the words closer and further, closest and furthest. Choose two volunteers
sitting clearly at different distances from the front of the classroom. Have them to stand
up. ASK: Who is closer to the front? Who is closer to the back? Who is further from the
front? Continue with distance from objects in the room. EXAMPLES: from the bookcase,
the teacher’s desk, filing cabinet. Then have four volunteers. Ask similar questions
regarding one volunteer’s position to another. EXAMPLES: Who is furthest from
Bonnie? Who is closer to Teah than Julia is?
EXTRA PRACTICE
BLM Closest
Determine which pair of dots is closer together. Draw the picture in the margin on
the board. Have students decide which pair of dots—above the line or below it—is closer
together. Remind students that “closer” means “more close.” Ask students to brainstorm
other words where the “er” ending means more (e.g. long and longer, fast and faster, hot
and hotter). Then continue having students decide which pair of dots is closer together
for various examples.
EXAMPLE:
Compare how close or far apart numbers are by looking at a number line. PSS –
Modelling Draw a number line from 0 to 6 on the board. ASK: Is 3 closer to 1 or to 2?
Draw dots at 1 and 3 above the number line and at 2 and 3 below the number line (see
the margin for an example).
Example: Is 3 closer to 1 or to 2?
0
1
2
3
4
5
Which number any number is closer to won’t depend on the number line drawn.
Draw two number lines from 0 to 6 that both compare how close 4 is to 1 with how
close 4 is to 2, but change the spacing of the numbers on the number line:
0
1
2
3
4
5
6
0
1
2
3
4
5
6
ASK: Is 4 closer to 1 or to 2 on this number line? (point to the first number line) How
about on this one? (point to the second number line) Explain that no matter how we
draw the number line, 4 is always closer to 2 than to 1. Mathematicians say that 4 is
closer to 2 than to 1 because that’s how it is on any number line.
All numbers on a number line must be the same distance apart. Draw a number line
where 0 to 7 are very close together and 7 to 10 are very far apart, so that 7 is closer to
0 than to 10.
01234567
8
9
10
ASK: Is 7 closer to 0 or to 10 on this number line? (to 0) Why did that happen? SAY:
When mathematicians say that 7 is closer to 10 than to 0, they mean that 7 will be closer
to 10 than to 0 on any number line, as long as all the numbers are the same distance
apart. Draw two different correct number lines, with all the numbers the same distance
apart, on the board to illustrate. On both of them, 7 is closer to 10 than to 0.
Closer to means fewer numbers away from. Have eight volunteers stand in line at
the front of the room, so that there is a clear front to the line-up. ASK: Is Jamie
closer to the front of the line or to the back? How do you know? How many people
are in front of him? Behind him? (Possible answer: More people are behind him than
in front of him, so he is closer to the front.) Is Wei closer to Hamide or to Mina?
(Possible answer: Wei is closer to Mina, because only one person is between him
and Mina but 3 people are between him and Hamide.)
Repeat with 11 volunteers each holding a large card from 0 to 10, to form a number
line. Instead of naming students, refer to the students by the number they are
holding. For example, ASK: Is the person holding 3 closer to the person holding 7 or
to the person holding 1? Is 8 closer to 5 or to 10? Emphasize that two numbers are
closer together if fewer numbers are between them.
EXTRA PRACTICE
BLM Closer to 0 or 10
Is the number closer to 0 or to 10? Provide each student with a number line from 0
to 10 (see BLM Number Lines 0 to 10). Ask students to determine which numbers
are closer to 10 than to 0 (6, 7, 8, 9), which numbers are closer to 0 than to 10 (1, 2,
3, 4), and which number is equally close to both. (5) ASK: Which numbers are more
than 5? (6, 7, 8, 9) Which number are they closer to—0 or 10? Repeat for numbers
less than 5.
Numbers with ones digit more than 5 are closer to the next ten. PSS Looking for a
pattern. Draw two number lines on the board, one from 0 to 10 and the other, right
underneath it, from 30 to 40. Or, if you have a pocket chart with 11 columns, so that the
tens appear on both the left and right sides, draw your students’ attention to it. ASK: Is
38 closer to 30 or 40? How do you know? (because 8 is closer to 10 than to 0, so 38 is
closer to 40 than to 30) Point out that since the ones digit is more than 5, the number is
closer to the higher 10. Repeat with more examples, and then have students do more
individually: EXAMPLES: Is 43 closer to 40 or 50? Is 76 closer to 70 or 80?
Determine the closest ten. Have students progress as follows. First, have students list
the numbers that are between two given tens (e.g. between 30 and 40); then, have
students find the tens that a given number is between. Volunteers may demonstrate
some of their answers by finding the number on a large hundreds chart (for example, a
pocket hundreds chart) and show the two tens it is between. EXAMPLE: What two tens
is 73 between? (70 and 80) Finally, have students determine the closest ten. Once you
know which two tens 73 is between, look at the ones digit, 3, to determine which ten it is
closest to—3 is less than 5, so 73 is closer to 70 than to 80. EXAMPLES: 49, 77, 12, 84
(see margin for example). Students can check their answers using measuring tape or a
metre stick.
EXAMPLE:
9 is ________ than 5
49 is between _____ and _____
49 is closest to _____
EXTRA PRACTICE
BLM Closer to 40 or 50
NS2-47
Estimating Numbers
Page xxx
Goals
Students will repeatedly guess and revise estimates based on grouping objects in
tens.
PRIOR KNOWLEDGE REQUIRED
can group objects in bundles of 10
can count objects to 10
can count by 10s
understands place value to tens and ones (e.g. 50 = 5 tens)
VOCABULARY
estimate
check
closest
array
about
MATERIALS
many straws cut in thirds
many pennies
many similar-sized beads
a full jar of jelly beans
an empty jar same size as jelly bean jar
cards with random dots (see below)
BLM Jelly Beans (p xxx)
BLM Quantity – 5 or 10
BLM Quantity – 10 or 20
CURRICULUM EXPECTATIONS
Ontario: 1m17, 2m1, 2m3, review
WNCP: 1N6, 2N6, [ME, V]
Estimating means guessing by using information. You will need the cards from
BLM Quantity – 5 or 10. Show students the back side of one of the cards and tell
students to guess whether the number of dots on the other side of the card is closer to
5 or 10. Now turn the card around for a short time and have them guess again
whether the number of dots is closer to 5 or 10. Finally, count the dots to decide
together if the number of dots is closer to 5 or 10. Repeat with the remaining cards.
ASK: Were your guesses more accurate when you saw the cards? SAY: When we
guess based on information instead of just wild guessing, we are estimating. Write the
word “estimate” on the board.
Repeat with the cards from BLM Quantity – 10 or 20, but this time tell students to
decide whether the number of dots is closer to 10 or 20. Again have students guess
blindly at first for each card, and then have them look quickly at the cards to estimate.
Review grouping objects to count them. For example:
There are 10, 20, 30, 31, 32, 33, 34, 35, 36, 37 dots. Students will need to do this on
Workbook p. 13. If students struggle with this, encourage them to write the tens above
the groups of ten as they count them.
Estimate how many by grouping 10. PSS – Guessing, checking and revising
EXAMPLE: Give each student a large handful of straws (cut in thirds) and have them
guess how many straw pieces they have. Take a bundle of 40 straws yourself and
model guessing 21. Have students bundle one group of 10 straws with elastics and
guess again. Model doing so yourself; explain how you know that 21 is not reasonable
anymore – there is a lot more than 10 left after bundling 10. Repeatedly bundle ten
straws and repeatedly revise the guesses.
SAY: You used more and more information as you made more guesses. Even your
first guess wasn’t a completely wild guess because you were using some information.
For example, no one guessed 3 straws because you could see that you had many
more than 3. So, you were always estimating, but your guesses got better and better
because each guess used more and more information.
Repeat with additional EXAMPLES: stacks of pennies; a pile of similar size beads.
Then show students a full jar of jelly beans (less than 100 jelly beans) and have
students guess how many jelly beans are in the jar. Record some of the initial
guesses on the board, and then have volunteers move 10 jelly beans at a time into a
second jar of the same size. After each group of jelly beans is moved, students should
revise their estimates. Keep track of how many groups of 10 have been removed and
when the two jars have about the same amount, tell students how many groups of 10
have been moved so far. ASK: How many groups of 10 do you think are left? Why?
(should be about the same because the jars look like they have the same amount of
jelly beans; students might guess one more or one less) How many groups of 10 do
you think were originally in the jar? (add the two numbers together, e.g 4 + 5 = 9
groups of 10 or 4 + 4 = 8 groups of 10) Then check together as a class how many
groups of 10. ASK: How many is that? EXAMPLE: 9 groups of 10 is 90 jelly beans
(check this with skip counting).
EXTRA PRACTICE
BLM Jelly Beans
Estimating to the closest 10. PSS – Mental Math, Visualization Hold up a card with
dots arranged randomly. EXAMPLE:
Have students estimate, to the closest 10, how many dots they think there are. ASK:
How many groups of 10 do you think there are? Circle a group of 10 and ask if anyone
wants to change their guess. Then circle another 10 and again ask if anyone wants to
change their guess. Continue in this way. ASK: Does showing a group of 10 make it
easier to estimate how many there are? Finish circling all groups of 10 and discuss how
grouping by 10 made it easier to count them all. Repeat with other dots on cards, but
group by 5 instead of 10.
Combining large dots and small dots. PSS Reflecting on what made the problem
easy or hard. Draw dots randomly on a card, similar to above, but this time draw
some large dots and some small dots. Have students estimate using the same
method as above. Group first a group of 10 small dots, and then a group of 10 big
dots. Discuss what made the number of dots harder to estimate accurately. (Because
the dots are not all the same size, it’s harder to get an idea for how much room 10
dots take up.) EXAMPLE:
Discuss what affects estimating. Size does. Does colour? (white or dark) Pattern?
(e.g. happy or sad face)
Then show students these lines:
ASK: Are they all the same size? (Yes) Are they easy to estimate how many? (no)
Why not – what makes them hard to estimate how many there are? (It’s like they’re
different sizes because they take up so much less space one way than the other.)
NS2-48
Even and Odd
Page xxx
Goals
Students will learn even and odd by pairing up groups made of even and odd objects.
PRIOR KNOWLEDGE REQUIRED
can count
VOCABULARY
pair up
divide
even
odd
equal
team(s)
MATERIALS
several pairs of socks (see below)
8 counters for each student
8 paper circles
CURRICULUM EXPECTATIONS
Ontario: 2m1, 2m5, 2m7, optional
WNCP: 2N2, [R, V, C, CN]
Introduce even and odd numbers by pairing socks. Show the students 2 identical red
socks, 3 identical blue socks and 4 identical green socks. SAY: I took these from the
dryer this morning and I think I have all of them. ASK: How can I check to make sure
none is missing? Have a volunteer help fold the socks in pairs. Write the word “pair” on
the board. SAY: Socks are worn two at a time, so we pair them up when putting them
away. ASK: Which colour of sock am I missing? How do you know?
Pairing up faces. Draw on the board:
SAY: I tried to pair up all the people, but one of them got left out. ASK: Can anyone
explain what I mean by pair up? (Group people into groups of two.) Write the words “pair
up” on the board. Give several examples of groups of happy faces and have volunteers
try to pair them up. Each time, ASK: Were you able to pair them all up or was one left
out?
Then draw on the board: PSS – Generalizing from examples
For each box, ASK: How many faces are there? Can you pair them all up without any
left over? If I have any eight faces, no matter how they are arranged, do you think that I
will always be able to pair them up? (This point is important: if we define a number as
being even when you can pair up objects, it must be true no matter how they are
arranged.)
Pairing up eight counters and introducing “even.” Place eight counters on an
overhead projector or stick paper circles on the board. ASK: Can I pair them up without
any left over? Then pair them up. Then give students eight counters each and tell them
to try to pair them up without any left over. ASK: Who was able to pair them all up? Who
was not? SAY: No matter how the counters are arranged, if you have eight of them you
will always be able to pair them up. Because of that, we say that eight is even. Write the
word “even” on the board.
Introduce the word “odd.” Draw seven happy faces on the board, arranged randomly.
ASK: How many happy faces did I draw? Have a volunteer try to pair up the faces.
ASK: Is seven even? If I line up the faces in a row, do you think I will be able to pair up
the faces? Then try it and show students that you cannot. SAY: No matter how you
arrange the seven faces, you will never be able to pair them up without any left over. So
seven is not even. Numbers that are not even are called odd. Write the word “odd” on
the board. Then draw several groups of stars on the board, and have students count the
number of stars and decide whether the number is even or odd by trying to pair up the
stars.
Use teams to determine if a number is even or odd. Connection – Real world
Arrange ten counters where students can see them, using five red and five yellow
counters. SAY: Let’s pretend the counters are people, one team has red jerseys and the
other team has yellow jerseys. Separate the red and the yellow counters and ASK: Do
these two teams have the same number of players? How can I tell without counting?
(pair up each red with a yellow) Demonstrate doing so. Are there an even number of
people altogether? (yes) How do you know? (because we could pair up the counters)
Repeat for other numbers, both odd and even. Emphasize that an even number of
people can always be divided up into two equal teams—an odd number of people
cannot be.
Draw pictures on the board such as:
Have students count the number of faces and decide from the picture whether that
number is even or odd. Show how to pair up the faces so that you use the definition of
even and odd:
SAY: Because there is one left over when we pair them up, there will be one left over
when we try to put them into equal teams. So 9 is odd. We could have teams that are
not equal with 5 on one team and 4 on the other, but we can’t have two equal teams.
NS2-49
Patterns with Even and Odd
Page xxx
Goals
Students will first discover that even and odd numbers alternate and then that the ones
digits of even and odd numbers can be used to identify them as even or odd.
PRIOR KNOWLEDGE REQUIRED
can pair up numbers
can check whether a number is even or odd
understands repeating patterns
understands the concept of equal teams
VOCABULARY
even/ odd
pair
pair up
core
repeating pattern
extend
alternate
ones digit
shaded
circle(d)
underline
before/ next
MATERIALS
BLM Even and Odd in a Hundreds Chart (p xxx)
CURRICULUM EXPECTATIONS
Ontario: 2m1, 2m5, 2m7, optional
WNCP: 2N2, [R, CN, C]
Look for a pattern of even and odd in consecutive numbers. Connection – Patterns
Do the first six questions on Workbook p. 17 together, then write on the board:
1
Odd
2
3
Even Odd
4
Even
5
Odd
6
7
8
9
10
Even _______ ________ ________ ________
ASK: Do you see a pattern in whether the numbers are even or odd? Is this a repeating
pattern? What is the core? (odd, even) By looking at the pattern, do you think 7 will be
even or odd? Repeat for 8, 9, 10. Have students verify their prediction by drawing
groups of 7, 8, 9 and 10 objects, and trying to pair them up.
Connect counting by 2 to saying the even numbers. Tell students that 2 is an even
number. ASK: What is the next even number? (4) And the next even number after 4? (6)
Point to the odd-even pattern above and explain that to find the next even number, they
skip one number and say the next. Say quietly, “skip 1” then loudly, “say 2”, then quietly:
“skip 3” and so on. ASK: What does this remind you of? (skip counting by 2s) Then write
on the board:
2
4
6
______
______
______
______
______
______ _____
Have students continue writing the even numbers up to 20 in their notebooks, by using
skip counting by 2s.
Then have students write just the ones digits of the numbers they found:
2 4 6 ______
______
______
______
______
______
______
ASK: PSS – Looking for a pattern Is this a repeating pattern? (yes) What is the core of
the pattern? (2, 4, 6, 8, 0) Have students extend the pattern of ones digits:
2
4
6
8
0
2
4
6
8
0
_____
_____ _____ _____ _____
Connect counting by 2s from 1 to saying the odd numbers. Repeat the exercises
above, this time starting with 1, to say the odd numbers.
Then ASK: What are the ones digits of the odd numbers? (1, 3, 5, 7, or 9) What are the
ones digits of the even numbers? (2, 4, 6, 8, or 0) Are these numbers even or odd?
EXAMPLES: 13 24 87 83 90 94 Bonus: 125 876 95 431
Write groups of numbers on the board. EXAMPLES: 7 8 9; 17 18 19; 97 98 99;
43 50 67; 5 10 15 20 25. Have volunteers circle the even numbers and underline
the odd numbers. Bonus: 657 789 031 8 967 540
Finding the next or previous even or odd number. Emphasize that students can skip
count forwards by 2s to say the next even or odd number and can skip count backwards
by 2s to say the even or odd number before a given even or odd number.
Is zero even or odd? Write “0” on the board. ASK: Is 0 even or odd? SAY: We cannot
pair up any objects if there are no objects to pair up, so it doesn’t make sense to say that
0 is even, but there isn’t a leftover object, so it doesn’t make sense to say that 0 is odd
either.
ASK: What is the ones digit of 0? (0) Does that fit with the even numbers or the odd
numbers? (even) Why? (because 0 can be the ones digit of an even number, but not of
an odd number, OR because the ones digits of even numbers are 2, 4, 6, 8, or 0) Now
write the following pattern on the board:
0
1
Odd
2
Even
3
Odd
4
Even
5
Odd
6
Even
7
Odd
8
Even
9
Odd
10
Even
ASK: What should 0 be to keep this pattern—even or odd? (even) Why? (because the
number next to it is odd) Explain that mathemticians call 0 even for two reasons—
because it fits with the repeating pattern, and because its ones digit fits with the ones
digits of even numbers, not the ones digits of odd numbers.
CONNECTION - Sorting
BLM Even & Odd in a Hundreds Chart
Extensions:
1. PSS – Making and investigating conjectures Have students add two odd numbers
together – what type of number do they always get? Start by providing examples for
them (3 + 5; 7 + 7; 5 + 1; 9 + 3; 1 + 7) then allow students to investigate by creating their
own examples. Bonus: Repeat with adding 3 odd numbers.
Show students why this works with counters. For example, 3 counters has one extra not
paired up and 5 counters has one extra so pair up the extras with each other.
3
+
+
5
=
8
=
2. BLM Equal Parts guides students to discover that 0 must be even in a different way
from how it was done in the lesson, this time using that even numbers are the sum of
two identical numbers and odd numbers are not.
3. PSS – Guessing, checking and revising, Using logical reasoning, Organizing data,
Working backwards To do BLM Even and Odd in Shapes, students only need to
know that 0 and 2 are even while 1 and 3 are odd. Still, the puzzle will require some
thinking and will be quite challenging.
NS2-50
Patterns in Adding
Page xxx
Goals
Students will discover ways to find all pairs of numbers that add to a given number.
Students will use pictures and concrete materials to model addition.
PRIOR KNOWLEDGE REQUIRED
can draw models to add pairs of numbers
can do missing addend problems
VOCABULARY
equal
first/ last
vertical line
addition sentence
pair
MATERIALS
many counters
one cup for each student
one large blank card per student
2-colour counters
number cards (see BLM Number Cards Template (p xxx))
CURRICULUM EXPECTATIONS
Ontario: 1m18, 1m25, 2m1, 2m5, 2m6, 2m7, review
WNCP: 1N4, review, [V, R, CN, C]
Write numbers in different ways. Write on the board: 3 + ____ = 7. Draw seven
circles, arranged randomly. ASK: How could you use the circles to find the answer?
PSS – Modelling (Possible answers: colour three circles and count how many are not
coloured; cross out three circles and count the ones that are left; circle a group of three
circles and count how many are not part of the group; and so on.)
Then draw a row of seven circles:
ASK: What’s an easy way to choose three circles? SAY: I could choose the first three or
the last three but I’m going to choose the first three circles. I find it easier to remember
that the first number in 3 + ___ = 7 goes with the first circles and the second number
goes with the second number of circles. ASK: How can we separate the first three
circles from the others? (colour them; cross them out; circle them) Explain that these are
all good ways, then show how to separate the circles by drawing a vertical line after the
first three and explain that this is the model the workbook uses:
ASK: Can you find the answer to 3 + ____ = 7 using this picture? How?
Now tell students that you want to find all the ways of writing 7 = ______ + ______. PSS
– Making an organized list Write “7 = ______ + ______” eight times on the board, all in a
vertical column. ASK: What is the smallest number that can be put in the first blank? Can
there be no circles before the line? (yes) Demonstrate this by drawing the line before the
first circle, and write 0 in the first blank. ASK: How many circles are after the line? (7)
Then finish the number sentence: 7 = 0 + 7. Continue in this fashion by asking what is
the next smallest number after 0? (1) Can there be 1 circle before the line? (yes) And so
on, until all 8 number sentences are complete. Point out how the line separating the
circles moves one to the right each time. ASK: Have we found all the pairs that add to 7?
(yes) How do you know that we didn’t miss any? (because we wrote the numbers in
order)
Discuss how the 8 number sentences relate to each other. PSS – Looking for a
pattern ASK: What number is the same in each addition sentence? (the total) How do
the other numbers change each time—what happens to the first number? (goes up by
one) What happens to the second number? (goes down by one)
Take 7 pennies and ask for a volunteer. Write on the board:
Volunteer’s pennies + My pennies = 7 pennies.
ASK: How many pennies does the volunteer have? (0) How many do I have? (7) How
many are there in total? (7). Write the number sentence (0 + 7 = 7) Now give the
volunteer a penny and repeat the questions. Emphasize that you did not change the total
number of pennies by giving one to the volunteer. But the number of your pennies went
down by 1 and the number of their pennies went up by 1. Repeat until all pennies are
transferred. Discuss how useful it is to be organized. By giving the volunteer one at a
time, you made sure you didn’t miss any numbers.
PSS – Making an organized list Write “8 = _____ + ______” nine times on the board,
and ask students for strategies to fill in the numbers with all possible answers. Some
students may suggest using a model or transferring pennies. If so, do so again. Then
SAY: Notice the pattern: from one addition sentence to the next, you add one to the first
number and subtract one from the second number, and you are not changing the total
number. Then challenge students to find all ways of writing 6 = ____ + ____ without
transferring pennies. Bonus: Find all ways of writing 13 = ____ + ____.
Activities 1-2
1. Counters in a Cup. Students move 1 counter at a time into a cup. Students write the
addition sentences based on Number in Cup + Number Not in Cup = Total Number.
2. Give each student a card to write a number sentence on. SAY: We want to find all the
pairs of numbers that add to 19 (or however many students are present.) All students
stand to begin. Write on the board two columns headed “Number Standing” and
“Number Sitting.” Write 19 + 0 = 19. Then have students sit down one at a time. As each
student sits down, the student writes the corresponding number sentence on their card.
When finished, collect all the cards and display them.
Are all the addition sentences needed? SAY: PSS – Using logical reasoning We
know that order doesn’t matter in addition. I see that some of the number sentences
have the same numbers. Have volunteers each erase one of each pair adding to 8 that
is the same. Repeat for pairs adding to 7.
Activity 3
Make a small pile of 2-colour counters. PSS – Visualization Count the total number of
counters together (say, 15). Then throw them up so that some land red and others land
yellow. SAY: I want to know how many landed on red and how many landed on yellow.
Cover them up and ASK: What are some possibilities? Write the corresponding number
sentences on the board. Then uncover and count to see if the actual amounts came up
in their list.
CONNECTION – Literature
One More Bunny by Rick Walton.
Students find many ways to add to numbers from 1 to 10 by using the pictures.
Domino addition by Lynette Long.
Students find pairs of dominoes that add to a given number.
NS2-51
Adding Tens and Ones
Page xxx
Goals
Students will write numbers as a sum of 10s and 1s.
PRIOR KNOWLEDGE REQUIRED
can add
knows the number of tens and ones in 2-digit numbers
VOCABULARY
ones digit
tens digit
sum
ones card
tens card
MATERIALS
9 tens blocks for each student
9 ones blocks for each student
BLM Hundreds Chart
BLM Tens Cards
BLM Ones Cards
CURRICULUM EXPECTATIONS
Ontario: 2m1, 2m7, 2m13
WNCP: 2N4, 2N7, [R, C]
Write numbers as a sum of 10s and 1s. PSS – Modelling, Looking for a pattern
Provide each student with 9 tens blocks, 9 ones blocks, and a hundreds chart that fits
tens and ones blocks (e.g. from BLM Hundreds Chart). Ask students to show 32 on the
hundreds chart using tens and ones blocks. SAY: Each tens block represents 10 and
each ones block represents 1 (count the ten ones together in one of the tens blocks), so
we can write 32 = 10 + 10 + 10 + 1 + 1 (3 tens and 2 ones). Have students continue to
show various numbers using tens and ones blocks, at first with a hundreds chart and
then without. Then have students write the addition sentences involving tens and ones.
Finally, have students write numbers as tens and ones without using tens and ones
blocks.
What number am I thinking of? Have students find the number for:
a) 3 tens and 4 ones (34) b) 4 tens and 3 ones (43)
c) 7 tens and no ones (70)
d) no tens and 4 ones (4) e) 9 tens and no ones (90) f) no tens and 9 ones (9)
g) 10 + 10 + 1 + 1 + 1 + 1 (24)
h) 10 + 10 + 10 + 10 + 1 (41)
i) 10 + 10 + 10 + 10 + 10 + 10 (60)
j) 1 + 1 + 1 + 1 (4)
Break numbers into their tens and ones. ASK: How many tens are in 35? (3) What
number is 3 tens? (30) How many ones are left? (5) Write 35 = 30 + 5. Have students
write various numbers as a sum of tens and ones. EXAMPLE: 42 (=40 + 2)
Adding tens is like adding ones. PSS – Changing into a known problem Have
students work in partners again and one partner has 9 ones blocks and the other has 9
tens blocks. Tell one partner to add 5 + 2 by grouping five ones blocks with two ones
blocks and finding out how many ones blocks they have altogether. Tell the other partner
to add 50 + 20 by grouping five tens blocks with two tens blocks and finding out how
many tens blocks they have altogether. Ask them how many ones did the ones person
have and how many tens did the tens person have. Are the answers the same? Why?
Emphasize that they can find 50 + 20 by counting the number of tens (5 + 2):
50 + 20 =
5 tens
+
2 tens
= 10 + 10 + 10 + 10 + 10
+
10 + 10
= 7 tens
= 70
Repeat with several examples—have students write out the tens in each number, and
then see how many tens they have altogether. EXAMPLES: 30 + 10; 40 + 20; 20 + 50;
30 + 30; 40 + 30. Now have students do similar problems without writing out the tens in
each problem:
50 + 40 = 5 tens + 4 tens
= 9 tens
= 90
Activity:
A card trick for adding tens and ones. Photocopy BLM Tens Cards onto blue paper,
once for each student, and photocopy BLM Ones Cards onto red paper, once for each
pair of students. Cut them out and give each student one set of blue cards (10 to 90) and
ones set of red cards (1 to 9). Show students how to add 30 + 4 using the cards. Find
the blue card 30 and the red card 4. Then place the 4 over the 0 on the tens card 30.
What number do they see? (34). Repeat with various other EXAMPLES: 20 + 7; 40 + 5;
80 + 3; 30 + 8. Have students hold up their answers. Discuss why this works to add tens
and ones. By covering up the zero with the ones digit, you are showing the number of
tens beside the number of ones. This is how we write numbers.
Now have students go in the other direction. Have students show the two cards that they
need to make various numbers. EXAMPLES: 73 (70 and 3), 84 (80 and 4), 48 (40 and
8). Students can then write the corresponding addition sentences. EXAMPLE: 73 = 70 +
3.
Extensions:
1. Show how to subtract tens. For example, to calculate 50 − 20, write 50 = 10 + 10 +
10 + 10 + 10, then cross out 2 tens; that leaves 3 tens, so 50 − 20 = 30.
2. Show how to add hundreds. SAY: Just like 10 is short for 1 + 1 + 1 + 1 + 1 + 1 + 1 +
1 + 1 + 1, 100 is short for 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10. The number
200 means 100 + 100. ASK: What does 300 mean? 500? 800? Continue with adding
hundreds. EXAMPLE: 500 + 300.
3. Have students do the BLMs “Switching Ones” and “Switching Tens.” These sheets
teach the children an application of separating the tens and ones digits to addition. It is
an extension of the commutative law. For example: 13 + 5 is the same as 15 + 3
because 3 + 5 = 5 + 3. Furthermore, by switching the tens, students will see that 36 + 20
= 26 + 30 because 3 + 2 = 2 + 3 and so 30 + 20 = 20 + 30. For extra bonus questions,
you can provide students with questions of the form:
46 + 32 = 36 + __ __ or 34 + 25 = 35 + __ __.
Online Guide: More Extensions
NS2-52
Adding in Two Ways
Page xxx
Goals
Students will use rows and columns to find the same total.
PRIOR KNOWLEDGE REQUIRED
understands quantity
knows addition facts
VOCABULARY
column/ row
separate
different
addition sentence
shaded
altogether
total number
MATERIALS
connecting cubes
BLM 2-cm Grid Paper
BLM Hanji Puzzles (p xxx)
many counters
3 toothpicks for each student
CURRICULUM EXPECTATIONS
Ontario: 1m18, 2m1, 2m3, 2m5, 2m13, 2m22
WNCP: 2N4, [CN, V, R]
Review that two different addition sentences can represent the same number.
Have students draw 2 rows of dots, each row with 7 dots. Have students separate the
first row with a line between two of the dots and then write a number sentence for the
model. Then have students separate the second row in a different place and write a
different number sentence. Then show students how to change the two sentences into
one addition sentence. EXAMPLE: 3 + 4 = 7 and 2 + 5 = 7 becomes 3 + 4 = 2 + 5.
Repeat with two rows of eight dots.
Bonus: Add 3 ways. EXAMPLE: 3 + 4 = 2 + 5 = 1 + 6
Count by rows to write addition sentences. PSS - Modelling Ensure that all students
have a clear understanding of the words “row” and “column” by asking students to
identify a row and a column on a hundreds chart or the calendar. Write the words “row”
and “column” on the board. Then draw the grid shown in the margin. ASK: How many
squares are shaded in the first row? (Write 3 next to the first row.) How many squares
are shaded in the second row? (Write 2 next to the second row.) How many squares are
shaded in total? Write the sum (5).
Repeat by adding an additional row to the bottom. Have a volunteer count the shaded
squares and write the addition sentence. Have students write the addition sentences for
these grids:
3
2
2
1
4
2
+ 2
+ 0
6
6
+
2
6
Count by columns to write addition sentences. Draw the same examples as above.
Have students determine the number of shaded squares in each column and ask them
to write the corresponding addition sentence as follows:
2+2+2=6
2+1+1+2= 6
2+2+2=6
Count by rows and columns to combine addition sentences. Have students
compare the row sum and column sum for each drawing. ASK: Are the numbers being
added always the same? (sometimes the column numbers differ from the row numbers)
Are the totals still the same? (yes) Finally, combine the two ways of finding the sums
(see margin).
3
1
+2
2 + 2 + 2 = 6
Repeat with the same examples as above. Have volunteers draw their own grids and
invite others to write the corresponding addition sentences.
Activity 1:
Have students work in groups of 3. First, each student individually makes a 3 by 4 grid
on grid paper. Assign each group a number of squares to colour, either 6, 7, or 8. Then
they pass their sheet to the person to the right and that person writes the addition
sentence from the row sums. Then they pass their sheet again and the next person
writes the addition sentence from the column sums. All students with 6 coloured work
together to make a “6” poster of all the ways they found to make a sum of 6. They cut
their grids and number sentences out and paste them to a common poster. Same with
the groups colouring 7 or 8 squares.
Compare the two models (rows of dots versus grids). PSS – Reflecting on other
ways to solve a problem Discuss why the grid model appears to provide more examples
of addition sentences than the dots model. (The grids allows for more than 2 numbers to
be added). Challenge students to find a way to use more than 2 addends with the dots
model. (draw more than one line to separate the dots in separate places) Provide
students with counters and toothpicks to do this concretely.
EXAMPLE:
2+5+5=3+3+3+3
Activities 2-3
2. Connecting cubes. Use 3 colours to write different number sentences with 3
addends.
R R B B B B Y
2+4+1=7
Use 2 colours to make number sentences with more than 2 addends by alternating
colours.
Y Y B B B Y B B
2+3+1+2=8
Then give each student 4 red counters and 1 yellow counter. Challenge them to
rearrange the counters to find many other addition sentences. Point out that the sum is
always 5 because you gave them 5 counters to begin with.
3. Cooperative Cards. Play the 2-player cooperative card game described online in an
At Home letter. Allow students time to discover the strategy on their own before sending
the game rules home with them. Repeat the game after they learn more addition
strategies.
AT HOME A cooperative card game and literature connections.
Extensions
1. Ask students to find 3 numbers that add to 7 in as many ways as possible without
using a model.
2. BLM Hanji Puzzles 1-3. The popular Hanji puzzles invert the exercises done in class;
instead of counting and adding the shaded squares in each row and column, students
are given the number to be shaded in each row and column. It is easier to start by
shading the full rows or columns.
3. Present the illustration shown in the margin. SAY: A student throws 3 darts. Each
lands on the board. ASK: What might the total score be?
1
2
5
6
9
7
3
ONLINE GUIDE Extension
The associative law: (2 + 3) + 4 = 2 + ( 3+ 4)
8
4
NS2-53
Addition Strategies
Page xxx
Goals
Students will be able to choose from a number of strategies that make adding easier.
PRIOR KNOWLEDGE REQUIRED
can write different models for the same sum
VOCABULARY
change
right
left
first/ second
opposite
MATERIALS
up to 20 counters for each student
toothpicks
a straw
CURRICULUM EXPECTATIONS
Ontario: 2m1, 2m2, 2m7, 2m13, 2m22
WNCP: 1N9, 2N4, 2N9, [C, V, R]
Adding 1 to the first number and subtracting 1 from the second number doesn’t
change the sum. PSS – Make an organized list Draw a row of seven dots with a line
after the second dot; use a straw taped to the board as the separating line.
ASK: What addition does this show? (2 + 5 = 7) Explain that there are 2 dots before
the line, 5 dots after the line and 7 dots altogether. Tell students that you would like
to move the line so that it shows 3 + ____ = _____. ASK: Which way should I move
the line – left or right (or say “this way or that way” while pointing)? Have a volunteer
move the line. Explain that you need to move it one dot to the right (this way) so that
there is one more dot before the line than there was.
SAY: There is now one more dot before the line – how did the number of dots after
the line change? (it went down by 1) Did the total number of dots go up, go down, or
stay the same? (it stayed the same) PROMPT: Did we add or take away any dots by
moving the line? (no) What will the new number sentence be? (3 + 4 = 7) Emphasize
how the number sentence changed.
2 +
5
= 7
+1
-1
3
+
4
= 7
PSS – Making and investigating conjectures Challenge students to predict what the
new number sentence will be when you move the line one dot to the right another
time. ASK: Does the first number go up by 1 or down by 1? Will there be more dots
before the line or less? (there will be one more dot before the line, so the first
number goes up by 1) Demonstrate this:
3
+1
4
+
+
4
-1
3
= 7
= 7
Continue moving the line one dot to the right, and emphasizing how the first number
goes up by 1 and the second number goes down by 1. Then have students repeat
the process with 8 dots. Start with:
Students should record the number sentences at each stage.
Practice finding another number sentence with the same answer. PSS –
Modelling Write across the board: 6 + 5 = 11. SAY: If I add one to the first number
and subtract one from the second number, I will still have a total of 11. Under number
6, write +1; under number 5, write -1, as shown. Complete the calculation with the
new number sentence: 7 + 4 = 11.
6 + 5 = 11
+1 -1
7 + 4 = 11
Have a volunteer draw the model to show what is happening.
Have volunteers continue with EXAMPLES: 8 + 9 = 17; 4 + 11 = 15; 5 + 12 = 17; first
writing a new number sentence and then drawing a model. Help volunteers at first by
inserting +1 and -1 under the addends, but eventually have them do this step
themselves. Finally, give each student up to 20 counters and a toothpick. Have them
do the steps concretely and then draw them pictorially and symbolically with addition
sentences.
Add 1 to the second number and subtract 1 from the first number. Repeat the
lesson, but this time, start by moving the line one place left in the same model and
discuss how each number changes or stays the same.
Doing the opposite to two numbers leaves their sum the same. Repeat the
lesson, but this time, start by moving the line two or more places left or right in the
same model and discuss how each number changes or stays the same.
Have students change both numbers in opposite ways to make another number
sentence with the same sum. Tell students how to change the first number and have
them decide the correct way to change the second number, so that the sum stays the
same. Students should check their answer by finding both sums.
Examples:1.
2.
5
-3
+
5
+3
+
4 = 9
4 = 9
Extension: What would
same?
3 + 4
+2
+1
5 + 5
5
+3
8
+
+
4 = 9
-3
1 = 9
5
+ 4 = 9
-3
+3
2
+ 7 = 9
you take away from the third number to keep the sum the
+ 5
= 12 (Answer: 2 + 1 = 3. Note that 5 – 3 = 2
and 5 + 5 + 2 = 12)
+ ____ = 12
NS2-54
Using 10 to Add
Page xxx
Goals
Students will use pairs of numbers that add to 10 to make adding easier.
PRIOR KNOWLEDGE REQUIRED
knows pairs of numbers that add to 10
can complete addition problems when one addend is missing
can add 10 to a 1- or 2-digit number
VOCABULARY
make 10
group
easier
MATERIALS
20 two-colour counters or coins
3 counters for each student
cards numbered 1 to 10 (3 or 4 of each number),
BLM Addition Table (Ordered) (p xxx)
BLM Cubes (p xxx)
BLM Pass the Puck (p xxx)
several shoebox lids
CURRICULUM EXPECTATIONS
Ontario: 1m26, 2m1, 2m3, 2m4, 2m7, 2m13, 2m22
WNCP: 1N10, 2N10, [C, R, CN]
Adding 10 is easier than adding 1-digit numbers. Have students add these sums in
their heads (students can use counting on their fingers):
7+8
8+9
6+8
5+6
7+5
Hide the answers and have students add these sums in their heads (again, students can
use counting on their fingers if it helps):
10 + 5
10 + 7
10 + 4
10 + 1
10 + 2
Discuss why adding 10 is easier than adding 1-digit numbers, even though adding
bigger numbers is usually harder than adding smaller numbers. PSS- Reflecting on what
made a problem easy or hard. Be sure to note the pattern—to add 10 to a 1-digit
number, just write the “teen” that ends with that number. Another way to see it is to use
tens and ones; for example, 1 ten and 7 ones is 17.
Finding an easier problem with the same answer. Now bring students’ attention to
the fact that the answers in the bottom row of questions are the same as the answers in
the top row (for example, 7 + 8 = 10 + 5).
Then draw on the board a model for 7 + 8 by drawing a group of 7 and a group of 8.
Count the two groups together to verify that this is a model for 7 + 8. Count all the dots
together and then write 7 + 8 = 15.
Then draw a big circle around 10 of the dots (see margin). ASK: Now what number
sentence does this show? PROMPT: How many dots are in the circle? (10) How many
are not in the circle (5) How many dots are there altogether—did I change the number of
dots by circling some of them? (no, there are still 15 dots) Write 10 + 5 = 15.
Discuss which question is easier to answer without counting all the dots: 7 + 8 or 10 + 5?
Why? (10 + 5 because you don’t even have to count; you can just look and know it’s 15)
Do they have the same answer? How do you know? (yes, because we just circled a
group of 10 without adding or taking away any dots)
PSS – Changing into a known problem Since 10 + 5 is easier to do than 7 + 8, and we
know they have the same answer, we might as well only do the easier problem. Repeat
with the other four problems.
ASK: How can we decide what to add to 10 to make the answers the same as the
questions in the top row? PROMPT: To get 10 from 7, what do I have to do? Write 7 +
____ = 10, and have a student fill in the blank. Explain that to keep the answer the
same, we need to do the opposite to 8. We added 3, so now we have to subtract 3.
Write 8 – 3 = _____ and have a volunteer fill in the blank. Then write on the board:
7 + 8 = _____
+3
-3
10 + 5 = ______
Challenge students to solve these problems by subtracting 3 from the second number:
7 + 5 = 10 + _____ = ______
7 + 7 = 10 + ______ = ______
7 + 9 = 10 + _____ = ______
7 + 6 = 10 + ______ = ______
Review finding numbers that add to 10. Tell students that it is important to be able to
determine what makes 10 with the first number so that they can know what to subtract
from the second number. Then have students practice finding what makes 10 with each
number from 1 to 9.
Write on the board: 10 = 1 + ___; 10 = 2 + ___. Have volunteers fill in the blanks.
Continue the pattern of addition statements so that all the pairs adding to 10 are on the
board. Give each student a card from 1 to 10. Write a number on the board and have all
students who have the number making 10 with the number you wrote, hold up their
cards. Erase the answers on the board and repeat.
Picking pairs. Use a deck of cards with the face cards removed. Count to make sure
you have 40 cards. Two cards “match” if they add to 10.
Finding the number that adds to 10 with the first number. Have students finish these
number sentences by finding what they need to add to the first number to make 10 and
then subtract it from the second number:
a) 7 + 9 = 10 + _____
= _____
b) 7 + 6 = 10 + ______
= _____
c) 9 + 7 = 10 + _____
= _____
d) 8 + 6 = 10 + _____
= _____
e) 6 + 8 = 10 + ______
= _____
f) 6 + 7 = 10 + _____
= _____
d) 6 + 5 = 10 + _____
= _____
e) 9 + 6 = 10 + ______
= _____
f) 6 + 6 = 10 + _____
= _____
Making ten with the second number instead of with the first number. PSS –
Reflecting on what made a problem easy or hard ASK: Which questions above have the
same answer? Why did that happen? (Sample answer: 7 + 9 and 9 + 7 because you are
adding the same numbers) Which question was easier: 7 + 9 = 10 + ____ or 9 + 7 = 10
+ ____? Why? (for example, students might find 9 + 7 easier because it is easier to do 7
– 1 than 9 – 3)
Emphasize that it doesn’t matter whether they find what makes ten with the first or the
second number – they should just do what is easier. Have students add these by making
10 with the bigger number:
a) 6 + 9 = 10 + ___ b) 4 + 8 = 10 + _____
c) 9 + 5 = 10 + _____ d) 7 + 6 = 10 + ____
Then try the same problems by making 10 with the smaller number. Which way is
easier? (making 10 with the bigger number because then you have to subtract less)
PSS – Changing into a known problem Explain that by changing one of the numbers
to 10, the problem becomes an easier problem with the same answer. Changing a
problem into an easier problem with the same answer is a strategy that
mathematicians often use to solve problems.
Adding 2-digit numbers by using tens. Then write on the board: 29 + 7. ASK: How
can we make this question easier? Work through the answer with the class using the
same notation as seen on Workbook p. 35. Add 1 to 29 and subtract one from 7 (see
margin). Have a volunteer complete the new addition sentence: 30 + 6 = ___. SAY:
Notice that 29 + 7 = 30 + 6 = 36. Repeat by adding more 2-digit numbers ending in 9
to 1-digit numbers. EXAMPLES: 39 + 5; 79 + 9; 89 + 4.
29
+1
30
7
-1
6
30 + 6 = 36 so 29 + 7 = 36.
Now write on the board: 32 + 9 = 40 + ____
and 32 + 9 = ____ + 10
Discuss which way is easier to do the problem. Do they get the same answer both
ways? If not, decide which is correct by counting up.
PSS – Selecting tools and strategies Now include problems that involve changing 8 to
10 (EXAMPLE: 45 + 8 = 43 + 10), and where both numbers have 2 digits (EXAMPLES:
37 + 19 or 46 + 28 or 29 + 36). Have students decide which number to change to a ten.
Activities 1-2 Adapted from A Companion Resource for Grade Two Mathematics by
Saskatchewan Learning.
1. Add on a 9 × 9 grid. Give each student a copy of the BLM Addition Table (Ordered)
and a small counter. Have the students toss their counters and write the answers to the
addition: if their counter lands on the column numbered 4 and the row numbered 9, they
write the answer to 4 + 9 in that square. In this way, students randomly generate
questions for themselves.
2. Make Egg Carton Dice. Make your own “dice” using BLM Cubes or have students
bring in 6-pack egg cartons (bring in extras in case some students forget). Start
collecting them several weeks before doing the activity. A 12-pack cut in half will also
work. To make the dice, have students write different numbers in each hole in the carton
or write the numbers on paper first and tape or glue them to the carton. Have them put
two counters into the carton and shake.
Students roll (shake) the dice and add. Make sure that when students shake the dice,
they cover up any holes where coins can fall out. Students play with a partner; if they roll
the same total, they get a point. Players can keep track of their scores using tallies if
they are familiar with tallies and counting by 5s.
To make the game harder, students can write the numbers 4, 5, 6, 7, 8, and 9 instead of
1 through 6. Students roll (shake) the dice and add. Students play with a partner; if they
roll the same total, they get a point. Players can keep track of their scores using tallies if
they are familiar with tallies and counting by 5s.
Variations:
1. Write the numbers 4, 5, 6, 7, 8, and 9 instead of 1 through 6 on the dice.
2. Use a 12-egg carton to imitate 12-sided dice.
3. Put three coins in the egg carton to imitate rolling three dice.
3. Pass the Puck. Provide each pair of students with twelve 2-colour counters or coins
(heads and tails act as two colours), a token to be the puck, and BLM Pass the Puck and
a shoebox lid. Each pair of students place their common puck at the start position.
Player 1 needs to toss some counters and move to an adjacent square in the next row
that shows how many of each colour turned up. Players can toss their counters into a
shoebox lid so that the counters do not fly across the room. Note that the start position is
adjacent to all squares in the top row. Since all the adjacent squares add to 11, Player
1’s best move would be to toss 11 counters. If there is no square that they can move to,
they can roll as many times as they need to. Demonstrate by “rolling” five red and two
yellow, so both 5 + 2 = 7 and 2 + 5 = 7 work. ASK: Can I move if I’m on the starting point
and roll this? Are there any addition sentences that add to 7 that I could move the puck
to, from the starting point? (no) Why not? What do all the number sentences add to?
(they all add to 11) So how many counters should I have rolled? (11) Demonstrate doing
this and moving to the appropriate spot. Explain that if there is no spot they can move to,
then they can toss the counters enough times so that they find a place to move to. The
goal is to get the puck into the net by tossing the counters at most 10 times. Play the
game once through by asking volunteers each time to tell you how many counters to roll.
Students can play the game repeatedly, taking turns who starts. The game is structured
so that regardless of which player starts, the other player will finish the game.
ONLINE GUIDE More Extensions
Extension
1. BLM Using 10 to Add guides students to add using 10 in a different way. For
example, to find 6 + 7, write 7 as 4 + something because 6 + 4 = 10. So 6 + 7 = 6 + 4 + 3
= 10 + 3 = 13.
NS2-55
Using Tens and Ones to Add
Page xxx
Goals
Students will add by separating the tens and ones by drawing tens and ones blocks
and using a tens and ones chart.
PRIOR KNOWLEDGE REQUIRED
knows how to add tens and ones
addition facts adding to 9 or less
VOCABULARY
tens
ones
altogether
separate
MATERIALS
tens and ones blocks
opaque bags
CURRICULUM EXPECTATIONS
Ontario: 2m1, 2m5, 2m6, 2m7, 2m26
WNCP: optional, [CN, R, V, C]
NOTE: If students do not know their addition facts adding to 9 or less, they will be
frustrated trying to add 2-digit numbers. Start with small tens and ones digits.
Use tens and ones blocks to add 2-digit numbers without regrouping. Give
students tens and ones blocks. Write “16” on the board and have students make 16
using tens and ones blocks. Tell them to set aside those blocks into a pile and then
to make 13 with more blocks – students should make a separate pile. Write on the
board: 16 + 13 = ____. Tell students to combine the two piles. ASK: How many tens
blocks do you have? (Write “2 tens blocks” on the board.) How many ones blocks do
you have? (Count them together and write “9 ones blocks” on the board.) What
number do you have in total? To guide students, write on the board: 2 tens + 9 ones
= _____. (29)
Show how to draw tens and ones blocks to represent the numbers (one tens block
and six ones blocks for 16, and one tens block and three ones blocks for 13). To
draw a tens block, draw ten small squares is an row. Students can then count the
small squares to verify their addition. Emphasize that now we can count the tens and
ones blocks from the drawing; we don’t need to actually have tens and ones blocks.
Again, there are 2 tens and 9 ones, so 16 + 13 = 29. Count the small squares in the
drawing to verify the addition.
Have volunteers repeat drawing tens and ones blocks to add.
12 + 14 = ____;
EXAMPLES: 17 + 12 = ___; 11 + 16 = ___;
13 + 16.
PSS – Modelling Then show students an easier, but more abstract, way to draw tens
blocks. Instead of drawing ten small squares in a row, draw just one long thin
rectangle. Emphasize that you find it too much work to draw the ten small squares.
For example, to draw a number like 32, it is much easier to draw:
than
Have students practice drawing several numbers using these simpler blocks.
EXAMPLES: 25; 52, 37, 73.
Then have students add using these simpler drawings. EXAMPLES: 24 + 33 = ___;
41 + 27 = ___.
Discuss the advantages and disadvantages: it is easier to draw but harder to verify if
you’re correct because you can no longer count the small squares.
Activity:
Give each pair of students 9 tens blocks and 9 ones blocks in an opaque bag. One
partner shakes the bag and then reaches in and blindly picks out 7 blocks. The other
partner takes the remaining blocks. Have students write the numbers they get
individually and then together add to find the total. Students repeat the process by
switching roles. Then ask volunteers to write down their addition statements they
found. Discuss the results as a class. ASK: Why is everyone getting a total of 99?
Why are so many addition statements different? Who had more blocks – the person
who chose blindly or their partner? (the partner) Who had the greater number? Did
the person with more blocks always get a bigger number? If so, challenge them to
find a way so that the person with more blocks has a smaller number. If not, ASK:
Why not? (a tens block has more small squares than a ones block so counting the
number of blocks doesn’t tell you the number of small squares)
Review separating tens and ones, then adding tens and ones. Have students
show 34 with tens and ones blocks. Have a volunteer draw the tens and ones blocks
on the board. Then tell students that it is convenient to group the tens and ones
separately. Group the 3 tens blocks from 34 and ASK: What number does this show?
(30) Group the 4 ones blocks from 34 and ASK: What number does this show? (4)
Write: 34 = 30 + 4. Repeat with 52 = 50 + 2.
Have students write various numbers as sums of tens and ones. EXAMPLES:
58 = ____ + ____ (50 + 8); 62 = ____ + ____ (60 + 2); 73 = ____ + ____ (70 + 3)
Then have students add tens to ones. EXAMPLES: 30 + 7 = _____; 20 + 3 = _____;
Add by separating the tens and ones (no regrouping). Write on the board:
34 = 30 + 4
+52 = 50 + 2
ASK: How many tens are there in 34? In 52? Altogether? (8) Write 80 underneath the
30 + 50 and explain that 3 tens + 5 tens is 8 tens = 80. How many ones are there? (4
+ 2 = 6) SAY: There are 8 tens and 6 ones. Write: 80 + 6 = ____. ASK: What number
is that? (86)
Repeat by asking volunteers to separate the tens and ones. Then have them add the
tens and add the ones and finally combine the added tens and ones to answer the
question. EXAMPLES: 43 + 54; 18 + 71; 23 + 42 + 14; 31 + 22 + 35.
Bonus: Add 4 or more numbers. EXAMPLES: 21 + 23 + 31 + 14; 22 + 41 + 14 + 11
+ 11.
Connect the two methods. PSS – Connecting Discuss how writing 34 = 30 + 4 is
the same as drawing 3 tens blocks and 4 ones blocks (the 3 tens blocks represent 30
and the 4 ones blocks represent 4) and how it is different (it is less writing to write 30
+ 4 than to draw blocks)
Add using a tens and ones chart without regrouping. PSS – Making a table/ chart
Instead of writing 34 as 30 + 4, now write it as 3 tens + 4 ones and show students
how to fill in a tens and ones chart. Explain that we can add the tens and ones
separately, just as we did before.
tens
3
5
ones
4
2
SAY: Now we can add the ones and tens. ASK: How many ones do we have? (6)
How many tens? (8) Complete the chart by filling in the total. Have volunteers fill in
additional charts with EXAMPLES: 41 + 28; 55 + 32; 73 + 22.
Then write the addition question inside the chart, as shown, and have a volunteer
write in the total:
tens
ones
4
7
1
1
Repeat with questions written inside the chart. EXAMPLES: 18 + 61; 37 + 22; 43 +
53.
Conclude by writing addition questions in vertical form without a chart, but use grid
paper format. Have volunteers answer the following EXAMPLES:
34
+ 52
64
+34
83
+13
23
+72
Then have students individually solve similar problems.
NS2-56
Regrouping
Page xxx
Goals
Students will add two digit numbers by regrouping 10 ones for 1 ten.
PRIOR KNOWLEDGE REQUIRED
can decompose 2-digit numbers into tens and ones
knows the tens digit as number of tens and ones digit as number of ones
can find the number that makes 10 with a given number
can add and subtract 10
can add single-digit numbers up to 9 + 9
VOCABULARY
trade
regroup
ones/ tens
chart
MATERIALS
BLM Addition Table (Ordered) (p xxx)
BLM Sum Cards (p xxx)
BLM Addition Table (Ordered Side) (p xxx)
BLM Addition Table (Unordered) (p xxx)
CURRICULUM EXPECTATIONS
Ontario: 2m1, 2m6, 2m7, 2m26
WNCP: optional, [R, V, C]
Addition facts to 9 + 9. Use Activity 3 from NS2-52 and Activity 1 from NS2-54 to
help consolidate the addition facts up to 9 + 9 = 18.
Review adding 1-digit numbers by regrouping to make 10. Write on the board: 7
+ 5 = 10 + ___. Then use ones blocks to represent the numbers 7 and 5. Move part
of the second pile to the first to make the number 10 as shown:
Line up 10 ones blocks to show they are equal to 1 tens block. Have students work
individually with additional EXAMPLES: 8 + 4 = 10 + __; 3 + 9 = __ + 10; and so on.
Add 2-digit numbers using tens and ones groups by regrouping. Demonstrate 27 +
19 with tens and ones blocks on the overhead projector as follows:
27
=
20
+
7
19
= 10
+
3
+
6
30
+
10
+
6
SAY: I am just rearranging the ones blocks that I have to add together to make a pile of
10. ASK: How does that make it easier to get the final answer? What is 27 + 19? (46)
Have students solve the following EXAMPLES: 27 + 38; 16 + 45; 53 + 39; 25 + 66.
Add without tens and ones blocks by separating the tens and ones. EXAMPLE:
46 = 40 + 6
+ 28 = 20 + 8
= 60 + 14
So 46 + 28 = 74.
More EXAMPLES: 37 + 28; 14 + 47; 52 + 38; 28 + 56.
Regroup tens and ones on charts. Show 3 tens blocks and 12 ones blocks and a tens
and ones chart. ASK: 3 tens and 4 ones is 32—is 3 tens and 12 ones 312? (No, the
number of tens and ones have to each be less than 10 to read the number this way.)
PROMPT: Do we say thirty-twelve? Remind students how we count up: 30, 31, 32, …,
38, 39, 40 (not thirty-ten) Then demonstrate trading 10 ones blocks for a tens block.
Have a volunteer fill out the next row of the chart with the new tens and ones.
Tens
3
Ones
12
Tens
Ones
3
12
4
2
Have students use charts to add these EXAMPLES: 20 + 26; 40 + 17; 50 + 22.
Use a tens and ones chart to add 2-digit numbers. SAY: Mathematicians like to turn
a harder problems, like adding 2-digit numbers, into easier ones, like adding 1-digit
numbers. Display a tens and ones chart beside base ten materials again, showing the
adding of the tens and ones:
27
+ 19
Tens
Ones
2
1
7
9
27 = 2 tens + 7 ones
+19 = 1 ten + 9 ones
3
4
16
6
3 tens + 16 ones
4 tens + 6 ones
Guide volunteers to fill in the appropriate boxes in the chart. Add the tens and ones first,
then regroup to find the answer.
Then write on the board: 54 + 28. Draw the blank chart again and ask volunteers to
show 54 and 28 using tens and ones blocks. ASK: Where do I put the number of tens in
54? How many tens are there in 54? And so on. Demonstrate filling in the first two
boxes. Then have volunteers fill in the remaining boxes. Repeat until all students are
comfortable. EXAMPLES: 36 + 37; 42 + 19; 17 + 44.
Activity
1. Solitaire. Practice single-digit addition, using BLM Addition Table 1 (Ordered) and
BLM Sum Cards. Cut out all the sum cards, shuffle, and have students fill the addition
table. When students have mastered this game, increase the difficulty by using BLM
Addition Table 2 (Ordered Side) or BLM Addition Table 3 (Unordered).
VARIATION: Set a limit on where the cards are played. The limit could be that the first
card can be played anywhere, but the second card must be played in a square that
touches (either a side or a corner of) a square already played.
2. Straws. Give each student a handful of straws cut in thirds. Have students bundle the
straws in tens to count how many they have. Then work in pairs and write an addition
sentence from totalling their straws. Students may need to regroup. Finally, work in
groups of four and write another addition sentence.
3. PSS – Making an organized list, reflecting on the reasonableness of an answer This is
similar to Activity 2, but give each student in a group of four, a different colour of straws,
say, red, blue, yellow and green. Have students make addition sentences based on two
colours. EXAMPLE: ____ red straws + ____ yellow straws = ____ straws. Tell students
to make sure they have each worked with every other member of their group, so they
should have a total of 6 addition sentences. Then have students make one addition
sentence with all four colours and teach them how they can use this to check their pairwise addition. First, have students line up the pairs that use the opposite colours:
____ red + ____ yellow = ____ straws
____ blue + ____ green = _____ straws
____ red + ____ blue = ____ straws
____ yellow + ____ green = ____ straws
____ red + ____ green = ____ straws
____ blue + ____ yellow = ____ straws
Point students’ attention to the first row. ASK: If I add the two totals, which straws am I
counting? Are there any missing? (no, none are missing because I am counting the red,
yellow, blue, and green straws) Then point students’ attention to the second row. ASK:
When I count these two totals, which straws am I counting? (all of them, too!) Repeat for
the third row. Emphasize that the totals in each row should be the same because you
are always counting all the straws. If the two sums in each row don’t always add to the
same number, they should look for their mistake!
ONLINE GUIDE
An extension
NS2-57
The Standard Algorithm for Addition
Page xxx
Goals
Students will learn the standard algorithm for addition.
PRIOR KNOWLEDGE REQUIRED
can add tens digits and ones digits
can model a number using tens and ones blocks
VOCABULARY
ones/ tens
column
regroup
standard algorithim
MATERIALS
dice for students
(or use the egg-carton dice students made)
BLM Make Up Your Own Cards (p xxx)
BLM Adding – Step 1 (p xxx)
CURRICULUM EXPECTATIONS
Ontario: 2m1, 2m3, 2m7, 2m26
WNCP: optional, 3N6, 3N8, [ME, R, C]
Review adding with tens and ones charts. Include regrouping. Draw four blank tens
and ones charts side by side, leaving plenty of room underneath, and have volunteers
complete tens and ones charts for these questions:
35 + 47;
56 + 24;
48 + 18;
27 + 69
Introduce the standard algorithm. PSS – Looking for a pattern Show the first problem
done using the standard algorithm, underneath the tens and ones chart.
1
35
+47
82
Lead a class discussion. First, ensure that students understand where the numbers
come from. ASK: How many ones are there in total? (12) How is that shown on the tens
and ones chart? (write the 12 under the 5 and 7) How is that shown in the new way? (the
1 is written above the tens and the 2 is written under the 5 ones and 7 ones) Explain that
1 is the tens digit and 2 is the ones digit, so it makes sense to write the 1 in the tens
column and the 2 in the ones column. ASK: How many tens are there in total? (8) How
do you get that from the tens and ones chart—what numbers did you add? (The 3 and
the 4) SAY: But 3 and 4 is only 7. How did I know to make it 8? (regroup 1 ten from the
12 ones, so add 7 + 1 = 8) How can you get the total number of tens from the new way
of adding? (The 1 ten from the 12 ones is already regrouped, because it is already with
the tens column, so we can add it right away: 1 + 3 + 4 = 8. Write down the standard
algorithm for 56 + 24 underneath the tens and ones chart for this sum, and go through a
similar line of questioning. Then have volunteers write down the new way of adding for
the remaining two sums; start them off by writing the question for them.
Write down the three steps:
1. Add the ones.
2. Add the tens.
3. Regroup ten ones for a ten if necessary.
PSS – Reflecting on other ways to solve a problem Note that this new way is just
combining the two steps of adding the tens and regrouping. It is just a shortcut for doing
the same thing. For example, when finding the number of tens in the sum, instead of
doing 3 + 4 = 7 and then 7 + 1 = 8, you can just do right away: 3 + 4 + 1 = 8. ASK:
Which two additions are being combined in the second question? (5 + 2 = 7 and 7 + 1 =
8 becomes 5 + 2 + 1 = 8) Repeat for the third and fourth questions. (4 + 1 = 5 and 5 + 1
= 6 becomes 4 + 1 + 1 = 6; 2 + 6 = 8 and 8 + 1 = 9 becomes 2 + 6 + 1 = 9)
Discuss why it is important to add the ones first--if they add the tens first, they may
forget to include the 1 extra ten that was traded for 10 ones.) SAY: It’s a bit tricky
because you have to add from right to left instead of from left to right. Many students
even in grades three and four sometimes have trouble remembering to add from right to
left because it is so different. That’s why it’s important to practice a lot.
What if some students are having trouble? Some students may be overwhelmed by
having to do both steps on Workbook p. 46. If this occurs, have them return to the first
page again. This time have students do the second step on the first page by writing the
number of tens in the grey box, after having just completed the first steps on all the
questions. Then they can try both steps at the same time on the second worksheet.
They can check their answers using the first page, since the two sheets have the same
questions.
EXTRA PRACTICE BLM Adding – Step 1
Estimating sums by using the closest ten. PSS – Mental math Have students guess
what the closest ten is to this sum: 28 + 41. Explain that they can use what they know is
the closest ten to each number being added—30 and 40. So it makes sense to guess
that 28 + 41 will be close to 30 + 40. ASK: What is 30 + 40? (70) Have students find the
actual sum. (69) Is 70 close to the right answer? (yes) Repeat with various sums.
EXAMPLES: 19 + 32, 43 + 21, 53 + 18.
ONLINE GUIDE Estimating activity
Activities 1-2
1. I Have —, Who Has —?. (See NS Part 1 – Introduction) Use BLM Make Up Your
Own Cards. Use 2-digit numbers on top (e.g. 28) and sums of 2-digit numbers on
bottom (e.g. 17 + 25). The student with the card described would say: I have 28, who
has 42?
2. Play Dominoes (See NS Part 1 – Introduction) with 2-digit numbers on one side and
sums of 2-digit numbers on the other.
ONLINE GUIDE Extension with BLM
NS2-58
Doubles
Page xxx
Goals
Students will use skip counting to double numbers. Students will use the double of 5 and
10 to double other numbers.
PRIOR KNOWLEDGE REQUIRED
can count
can skip count by 2s
knows the double of 5 is 10
can add 10 to a one-digit number
VOCABULARY
double
doubles sentence
skip counting
rows
symmetry
mirror
model
MATERIALS
10 counters for each student
aper counters
BLM What is the Double? (p xxx)
BLM Doubles 1-3 (p xxx)
CURRICULUM EXPECTATIONS
Ontario: 1m26, 2m1, 2m22
WNCP: 1N10, 2N10, [R, V]
Introduce “double.” Write the word “double” on the board. ASK: Does anyone know
what the word double means? (add the same number to the number you have) If I have
3 pennies and I double the number of pennies, how many will I have? Demonstrate
counting out 3 pennies and then 3 more. Explain that if you double a number, you add
the same number again. Tape paper counters to the board to demonstrate this. Show 2
counters and SAY: I’m going to double my counters, so if I start with 2, I need to add 2
more. (put 2 more counters on the board) ASK: How many do I have now? Write: 4 is
the double of 2. Repeat with other examples, always emphasizing the word double and
using pictures or concrete objects to illustrate the doubling.
Double a number by creating 2 rows of the same number. Put a row of 3 paper
counters on the board and write 3 beside it. Then add another row underneath and ASK:
How many are there now? Write “6 is the double of 3.” Repeat with doubling other
numbers from 1 to 10. Give students ten counters. Have students write addition
sentences to show the doubles of 4, 2, 5, 1, and 0.
Reading doubles from a chart. Draw 2 rows of 1, 2, 3, 4 and 5 to demonstrate
doubling (see margin).
1 doubled
2 doubled
…
is 2
is 4
Now, using a piece of paper, cover up part of the 2 rows of 5 to show 3 doubled
(emphasize that you are leaving 3 in each row uncovered):
ASK: What is 3 doubled? (6) Have a volunteer use the paper to show 2 doubled, then 4,
then 1, and finally 5 doubled. Now draw 2 rows of seven dots and have volunteers use it
to show 3 doubled and 6 doubled, then 5 doubled, and so on.
Skip counting by 2s to double. Next, demonstrate counting the number in the top row
(count by 1s) and the total number (count by 2s). See margin. ASK: When I count the
total number, in the two rows, how am I counting? (counting by 2s)
one row:
1
2
3 … 7
two rows:
2
4
6 … 14
PSS Drawing a picture. ASK: Can you tell from this picture what the double of 4 is?
Repeat for 6, 3, 7, 2, 5, and 1. Then demonstrate counting by 2s, using your fingers, to
double 3—count by 2s until you have 3 fingers up. Connect to the chart above: hold up
one finger at a time as you skip count and point to the number in the first row. Ask
several volunteers to find the doubles of numbers up to 10 using this method.
Use 5 to double. PSS Changing into a known problem. Tell students you want to double
the number 8 in a different way. Show 8 as 5 coloured circles and 3 blank circles. Then
double by drawing two rows. Emphasize that the 2 rows of 5 is 10 circles and the two
rows of 3 is 6 circles, so 8 doubled is 10 + 6 = 16 (see margin).
8 is 5 + 3, so 8 doubled
is 10 + 6 = 16.
ASK: Why is adding 10 + 6 easier than adding 8 + 8? (because adding 10 is always
easy) Explain that because 10 is the double of 5, it is easier to double numbers when we
split the number into 5 plus another number. Repeat the above model with additional
examples. Then double some numbers using just the numbers, without the model: 6 = 5
+ 1, so 6 + 6 = 10 + 2 = 12. Have students double more numbers this way.
Bonus: Find the double of 13. (use 13 = 5 + 8 or 5 + 5 + 3 to get 10 + 16 or 10 + 10 + 6
= 26)
EXTRA PRACTICE
BLM What’s the Double?
Students double the numbers from 0 to 9 without the model.
Extensions:
1. Teach students to double 2-digit numbers. See BLM Doubles 1-3.
ONLINE GUIDE Details for teaching this extension
2. BLM Big Cubes and Cm (p xxx). Students discover that the number of small cubes
is double the number of large cubes for any given length.
Connection – Measurement
3. Challenge students to double numbers in different ways, including by subtraction, and
verify that they get the same answer. EXAMPLE: 7 = 5 + 2 so 7 doubled is 10 + 4 = 14,
but 7 = 10 – 3 so 7 doubled is also 20 – 6 = 14.
NS2-59
Using Doubles to Add
Page xxx
Goals
Students will use doubles to add by using the concept of one more than or one less
than.
PRIOR KNOWLEDGE REQUIRED
knows the doubles of numbers to at least 10
can complete number sentences where one number is missing
can add 10 to a one-digit number
can solve addition sentences with three addends
can find one more than and one less than
VOCABULARY
double
more than/ less than
the same as
two ways
symmetry
mirror
MATERIALS
counters
BLM Adding with Doubles (p xxx)
Curriculum Expectations
Ontario: 1m26, 2m1, 2m3, 2m4, 2m7, 2m26
WNCP: 1N10, 2N10, [C, R, ME]
Using one more, one less than pairs adding to easy sums. Review pairs of numbers
that add to 10. Examples: 7 + ____ = 10; 6 + ____ = 10; 8 + ____ = 10
Then have students decide whether the sum is one more than or one less than 10 by
comparing to two numbers that add to 10:
5 + 6 is ______one more than _______ 5 + 5 so 5 + 6 = _____
5 + 6 is _________________________ 4 + 6 so 5 + 6 = _____
8 + 3 is _________________________ 8 + 2 so 8 + 3 = _____
8 + 3 is _________________________ 7 + 3 so 8 + 3 = _____
4 + 5 is _________________________ 4 + 6 so 4 + 5 = _____
4 + 5 is _________________________ 5 + 5 so 4 + 5 = _____
PSS – Reflecting on the reasonableness of an answer Point out that several questions
were done twice. ASK: Did you get the same answer both times? Should you get the
same answer both times? (yes) Discuss how useful it can be to do the same question
twice – it can help you to know if you made a mistake.
Now have students decide which pair of numbers adding to 10 they should use to add.
EXAMPLES: 9 + 2 (one more than 8 + 2 or 9 + 1, so 9 + 2 = 11)
4 + 5 (one less than 4 + 6 or 5 + 5, so 4 + 5 = 9)
Using one more, one less than doubles. PSS – Mental math Review finding the
double of numbers from 1 to 10. Then have students decide if the sum is one more or
one less than a given double.
5 + 6 is _________________________ 5 + 5 so 5 + 6 = _____
5 + 6 is _________________________ 6 + 6 so 5 + 6 = _____
Repeat with 4 + 5 (compare to both 4 + 4 and 5 + 5). Point out the questions that were
done twice, and again discuss the value of doing the same question twice. Then add 6 +
7 and 7 + 6 by comparing both to 6 + 6. ASK: Do these questions have the same
answer? (yes) How could you have predicted this? (they are adding the same numbers)
How many more or less than a double. Show students the example 6 + 9 is 3 more
than 6 + 6 = 12, so 6 + 9 = 15. OR 6 + 9 is 3 less than 9 + 9 = 18 so 6 + 9 = 15.
Start by having students find the double fore using it to add. EXAMPLES:
7 + 7 = ____ so 8 + 7 = ____
4 + 4 = ____ so 4 + 5 = ____
9 + 9 = ____ so 9 + 8 = ____
6 + 6 = ____ so 8 + 6 = ____
7 + 7 = ____ so 7 + 4 = ____
4 + 4 = ____ so 7 + 4 = ____
3 + 3 = ____ so 3 + 5 = ____
7 + 7 = ____ so 7 + 9 = ____
5 + 5 = ____ so 5 + 9 = ____
9 + 9 = ____ so 7 + 9 = ____
Bonus: 12 + 12 = ____ so 12 + 13 = ____;
14 + 14 = ___ so 14 + 11 = ____;
33 + 33 = ____ so 35 + 33 = ____;
123 + 123 = ____ so 123 + 125 = ____.
ASK: Which questions were done two ways? Did you get the same answer both ways?
Have students decide which double to solve before using it to add. EXAMPLES:
6 + 5 = ___ (use either 5 + 5 = 10 or 6 + 6 = 12);
7 + 6 = ___ (use either 6 + 6 = 12 or 7 + 7 = 14).
8 + 5 = ____
9 + 7 = _____
6 + 8 = _____
6 + 9 = _______
Bonus:
41 + 42 = _____;
60 + 63 = _____;
312 + 310 = _____;
2341 + 2344 = _____.
EXTRA PRACTICE
BLM Adding with Doubles
Do the opposite to both numbers to make a double. PSS – Changing into a known
problem Write 3 + 5 = ____ on the board, and ASK: Does this look like a double? (no)
Why not? (the numbers are not the same) Cut out circles to tape to the board and put 3
in one pile and 5 in another pile. Challenge students to move only one circle so that both
piles have the same number of circles. ASK: What double do you see? (4 + 4 = 8) Did
we change the total by moving one circle? (no) Explain that 3 + 5 is the same as 4 + 4,
so even though it doesn’t look like a double, we can still use doubles to find it.
Now explain that we are really adding 1 to the pile with 3 and removing 1 from the pile
with 5. So we are doing opposite things to both piles:
3
+1
4
5
-1
4
So 3 + 5 = 4 + 4. ASK: How can we get a double from 8 + 6? From 5 + 7?
Bonus: How can you get a double from 13 + 15?
= ___.
ONLINE GUIDE
Activity using MIRAs
Compare the different ways of adding. Write 6 + 7 = _____. Challenge the class to
come up with as many different strategies as they can to solve the question. You
could get them started with: Start at 6 and count on until you have 7 fingers up.
Demonstrate this by saying 6 with no fingers up and then 7 with one finger up, and so
on, until you have 7 fingers up. Other strategies include using 10 (6 + 7 = 10 + 3 = 13)
or using doubles (6 + 6 = 12 so 6 + 7 = 13) Discuss which way is easiest? Which way
is slowest? Emphasize that if students know their doubles they only have to add 1, so
this is the easiest. Using 10, although easier than counting on past 6, still requires
students to subtract from 7 – 4, since 4 makes 10 with 6 (or 6 – 3 if they use what
makes 10 with 7). Emphasize that, by doubling, students are changing the problem
into two simpler problems that they already know how to do, that is, doubling and
adding 1.
Choose between using 10 or using doubles. PSS Selecting tools and strategies.
EXAMPLES:
• 7 + 6 = ___ (This is one more than 6 + 6 and three more than 4 + 6 or 7 + 3. Some
students may think it’s easier to find pairs that add to 10 than to find doubles.)
• 7 + 4 = ___ (one more than 7 + 3 or three more than 4 + 4 or 3 less than 7 + 7)
• 5 + 6 = ___ (5 + 5 is both a double and a pair adding to 10, so it doesn’t matter
which way you look at this one)
• 32 + 33 (using 10 doesn’t make sense; double in this case)
Online extensions for NS2-51
1. Write two numbers, one above the other, either one apart or ten apart, and have
students circle the pair of digits that are different and write whether the top number is 1
more, 1 less, 10 more, or 10 less than the bottom number. Example:
26
26 is 1 less than 27.
36
36 is 10 more than 26.
27
26
More examples: 39, 49 (10 less); 68, 67 (1 more); 36, 26 (10 more); 40, 41 (1 less).
Do not include pairs of numbers with both digits different (e.g. not 39, 40)
2. Ask students to write down a number that uses the same number of tens blocks as
ones blocks. Have students compare their answers with other students. Did other
students get the same answer or different answers? Repeat for a number that …
… uses more tens blocks than ones blocks.
… uses more ones blocks than tens blocks.
… uses three more tens blocks than ones blocks – ask: does 30 work here?
… has a 3 in it and uses two more ones blocks than tens blocks.
… uses a total of nine blocks.
If students know even and odd numbers, you can ask them to find a number that …
… is an odd number that uses more than seven tens blocks.
… uses an even number of tens blocks and an odd number of ones blocks.
… uses an odd number of tens blocks and an even number of ones blocks.
… uses an odd number of tens blocks and an odd number of ones blocks.
… uses an even number of tens blocks and an even number of ones blocks.
… an odd number that uses a total of eight blocks.
4. Give students several tens and ones blocks. Have students make a triangle
using tens blocks as edges (sides) and ones blocks as vertices (corners).
What number do their tens and ones blocks represent? (33) Repeat with a
square (44) and a pentagon (55). Have students predict what number their
blocks will show if they make a hexagon. (66)
Online Extension for NS2-52
The Associative Law. NOTE: The Associative Law for addition means that when
adding three numbers, adding the sum of the first two to the third gives the same result
as adding the sum of the last two to the first. EXAMPLE: 2 + 3 + 4 may be added as (2 +
3) + 4 = 5 + 4 = 9 or 2 + (3 + 4) = 2 + 7 = 9.
Using connecting cubes, show students red, green, and blue cubes like this:
R R
G G G
B
ASK: How many cubes are there altogether? Then have a volunteer complete the
addition sentence: 6 = _____ + ______ + ______. SAY: There are three groups of
cubes, therefore we have three parts to the addition sentence. ASK: What would
happen if we put the first two groups together. What would the number sentence look
like then?
R
R G G G
B
Have a volunteer finish the number sentence: 6 = ___ + ____.
Then, pull the red and green cubes apart, and put them back into the original set. ASK:
Does it makes sense to write: 2 + 3 + 1 = 5 + 1? Is the sum on the left the same as the
sum on the right? Does it make sense to put an = sign between them? (yes) Did I
change the total number of cubes by moving the cube in the second group to the first
group? (no) Do you think you would change the number of cubes if you moved the
second group to the third group? What would number sentence look like? Have a
volunteer move the green cubes to join the blue cubes instead and write the number
sentence on the board. Then write on the board:
6=5+1
6=2+3+1
6=2+4
Repeat with several examples. Always make the connection between the connecting
cubes and the addition sentences. Encourage students to write the longer number
sentence. EXAMPLE: 2 + 3 + 1 = 5 + 1 = 2 + 4. Circle the grouped numbers to help
them:
2+3+1
=
2+3+1
=
2+3+1
=
2+3+1
2+3+1
=
2 + ____
=
____ + 1
=
______
Then draw several examples of “piles” of circles on the board. SAY: Imagine that the
circles are counters in piles. We can move the first two piles together or the second two
piles together. Demonstrate moving actual counters at the same time as drawing the
circles symbolically on the board.
Using ten counters, have students work in pairs to create a matching addition statement
with three addends. Then, have students regroup the counters to form two new twoaddend addition sentences which still have a sum of ten. Have one partner write the
number sentence obtained by grouping the first two piles together and the other partner
group the last two piles together. Challenge students to use the sums 11, 12, 13, 14 …
18 to create addition sentences which follow the Associative Law.
Provide students with BLM The Associative Law (p xxx). Ensure that they understand
that the middle pile is being moved to either the first pile or last pile. Then pull all the
piles together to find the total. Practice drawing what the piles would look like after being
moved and writing the addition sentence.
Online extensions for NS2-54
2. Which is quicker: the brain or a calculator?
Challenge students to add faster than a calculator. First give students problems that only
add to either 5 or 10, and tell them you are doing so. You solve all the problems by
punching them onto a calculator at normal speed, and the students solve all the
problems in their head. Did they finish ahead of you? (Only they need to know the
answer; do not ask students to tell the rest of the class whether or not they finished
ahead of you)
Then progress to some problems adding to 4, some to 5, and some to 6. Teach students
to determine what makes 5 with the first number and then decide whether the second
number is one less than, equal to, or one more than that number. Repeat with problems
that add to 3, 5, or 7 with 2 less than and 2 more than. Then progress to problems that
add to 9, 10, or 11, and then to 8, 10, or 12. Finally combine two or more types of
problems on the same handout.
3. Show students the following puzzle:
SAY: Suppose we have:
2
4
5
Explain that by drawing arrows on some of the lines you are showing which numbers are
to be moved from the squares into the circle to be added. The following examples
indicate how each number from the diagram above would move into the circle.
2
4
5
Draw 2 or 3 arrows to indicate that those 2 or 3 numbers are to be added and guide
students to find the answers:
6
11
(ANSWERS: 2 + 4 = 6 and 2 + 4 + 5 = 11)
Then challenge students to find:
(ANSWERS: 2 + 5 = 7 and 4 + 5 = 9)
Repeat with EXAMPLES:
1
2
3
2
3
4
3
5
6
Then draw the following and ASK: What do you think this means?
1
2
3
ANSWERS: 1 + 1
3+3
2+2
Have students compare the answers to the following questions for each example above:
1.
+
(The answers are 2 + 5 = 7
and
+
and
3 + 4 = 7)
The pictures for both answers are the same:
2.
+
4+4=8
+
5+3=8
answer picture
3.
+
6+3=9
+
4+5=9
answer picture
Students will see that the numbers are always the same and the pictures for the
answers should help them see why. BLM Addition Puzzles 1-3 (p xxx) provides
practice with this type of puzzle.
Online Extension for NS2-56
A different way to use tens and ones charts to add. SAY: I will still add the tens and
ones separately, but I am going to put the tens and ones in a different place this time. To
make it easier, let’s just start by adding the ones. We’ll add the tens later. Draw a tens
and ones chart on the board as follows:
Tens
2
Ones
7
1
9
1
6
ones
7 + 9 = 16
SAY: Remember, we are just considering the ones at this point. ASK: Why am I writing
1 in the tens column? Why do I have a 6 in the ones column? (7 + 9 = 16; 10 ones = 1
ten and is written in the tens column leaving 6 ones in the ones column)
Repeat with examples, asking students to add only the ones. Emphasize why you are
putting the tens digit in the tens column and the ones digit in the ones column. Then
extend the chart so that you can add the tens as well.
Tens
Ones
2
7
1
9
1
3
6
ones
tens
ASK: Why am I writing 3 tens in the same column as the 1 ten from the 16? Can you tell
easily how many tens are in 27 + 19 from this chart? Can you tell easily how many more
ones there are in 27 + 19? What is 27 + 19?
Do more examples. Have students find the ones first, then the tens and then the total.
Bring them to the point where they do not need the tens and ones charts to do the
adding. To start, for EXAMPLE:
Tens Ones
2
7
+ 1
1
9
6
3
4
6
Eventually move away from writing the tens and ones on top.
NOTE: This method can provide a good intermediary step before learning the standard
algorithm. It is important, however, not to replace it with the standard algorithm, as it will
not always work so efficiently with 3-digit numbers:
+
2
3
7
4
6
5
1
2
+
2
3
7
4
6
5
1
2
9
9
6
6
6
10
2
2
1
0
6
7
0
2
Notice that the algorithm used for 2-digits does not quite work for three digits when you
carry a 1 and the two tens digits add to 9. You need to add a bit of inefficiency to make it
work, as indicated above. Nonetheless, even when adding 3-digit numbers, some
students may prefer this method as a starting point.
Online Extension for NS2-57
2. Provide BLM More Addition Puzzles 1-5 (p xxx). The last page provides blank
puzzles. The following are sets of 4 numbers that you could put in the squares for
students (or allow students to choose their own): 2,3,6,4;
6,7,5,9;
18,7,6,15;
27,34,18,16;
23,16,17,35. Discuss the self-checking mechanism this provides.
Online Activity for NS2-57
Estimating Game. Students need the egg carton dice described in NS2-54: Using 10
to Add for this game. Draw on the board the picture in the margin.
Give each student four tokens to mimic rolling 4 dice in the egg carton. Students try to
place the numbers in the four boxes to obtain a number as close as possible to 100, and
record their answer. Repeat, using the same numbers, to obtain a number as close as
possible to 70, and then finally to 40. Students could use BLM Estimating Game, which
provides the outline for the boxes.
Variation: Students roll one die (or use one token) at a time. Students record the
number in a square after each roll and are not allowed to change their minds based
on their next roll.
Online Details for teaching Extension 1 of NS2-58
First, use ten to double in the same way we used 5 to double on Workbook p. 49.
Example: To double 13, fill in the blank:
13 = 10 + ____. Then double the 10 and the number in the blank: 13 = 10 + 3, so 13
doubled is 20 + 6 = 26. Emphasize that this is the same answer as in the bonus above.
No matter how you double 13, you still get the same answer. Have students practice
using 10 to double various numbers. EXAMPLES: 11, 14, 13. Bonus: 17, 18, 16, 19.
Next, double tens by doubling the first digit. 20 is two rows of 10, so 20 doubled is four
rows of ten. This means that 20 doubled is 40. Have students double 30 (60) and 40
(80). Bonus: Double 70 (140) and 50 (100).
Then double numbers by separating the tens and ones: 34 = 30 + 4, so 34 doubled is 60
+ 8 = 68. Repeat with more 2-digit numbers where both digits are less than 5.
EXAMPLES: 24, 21, 32, 33, 31, 42, 41, 43, 44, 34, 22, 23.
Online Activity for NS2-59
Explain that we see doubles in a mirror. We see the real object and then we see it again
in the mirror. Give students MIRAs. Tell students to place four counters in front of the
MIRA. Ask a volunteer to draw what they see. Have another volunteer write an addition
sentence based on the number of counters on each side of the MIRA and the number of
counters they see altogether. ASK: Do you see a double?
Using MIRAs, have students model doubles plus one, such as 4 + 4 + 1, using counters.
One counter will always be outside the range of the mirror, as shown. Provide students
with BLM Doubles and Mirrors.