Download UnderstandPatterns Lesson 7

Focus on Math Concepts
Lesson 7
Part 1: Introduction
CCSS
3.OA.D.9
Understand Patterns
What are patterns?
A pattern is something that repeats. Sometimes we see patterns in shapes or letters.
Other times numbers make a pattern. Patterns can also be found when we use
numbers to add, subtract, multiply, or divide.
You can also see patterns in things around you. Look at the line of kids below.
Think How can you describe a pattern?
What do you add to
get to the next
number in the
pattern? Telling about what repeats in a pattern is called the rule.
The rule for the line of kids is boy, boy, girl.
You can also use numbers to describe this pattern.
1
2
3
4
5
6
7
8
9
The numbers 3, 6, 9, . . . tell where the girls are in line. Because the numbers in the
pattern do not repeat, you need to look for something that is done over and over to
get from one number to the next number.
52
L7: Understand Patterns
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Part 1: Introduction
Lesson 7
Think How do you know what numbers come next in a pattern?
You can use the rule to figure out what other numbers are
in the pattern. To get from one number to the next in the
pattern, you add 3.
The pattern is continued in the chart below.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22 23
Putting the numbers
in a chart helps me
notice things I might
not see if I just made a
list of numbers.
24 25 26
You may also notice other things about the pattern that can help you figure out what
numbers come next.
The numbers form diagonals in the hundreds chart. You can use them to tell the next
number in the pattern is 27.
The numbers in this pattern also alternate: even, odd, even, odd. Since 27 is odd, you
know that the number that comes after 27 will be even.
Reflect
1 Write your own number pattern that has at least six numbers in it. Then, describe
the rule and one other thing you notice about the pattern.
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53
Part 2: Guided Instruction
Lesson 7
Explore It
Use the information below to help you think about patterns in addition.
Rick has a pack of 100 baseball cards and likes to sort them into 2 piles.
He notices that when he has a pile of 20 cards, the other pile has 80 cards.
When he has a pile of 30 cards, the other pile has 70. When he has a pile
of 40 cards, the other pile has 60. Finally, when he has a pile of 50 cards,
the other pile has 50 too.
2 Rick shaded 20 squares in the first grid to show how many baseball cards were
in the first set of piles he made. Shade the rest of the grids to show the other
sets of piles.
3 What do the shaded squares show in each grid?
4 What do the white squares show in each grid?
5 What happens to the number of shaded squares as you move from one grid
to the next? 6 What happens to the number of white squares as you move from one grid
to the next? 7 Describe the rule for this pattern.
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L7: Understand Patterns
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Part 2: Guided Instruction
Lesson 7
Talk About It
Solve the problems below as a group.
8 The grid on the left shows Rick’s piles of 30 and 70. The grid on the right shows
Rick’s piles of 40 and 60. Shade the grid in the middle to show what happens if
Rick put 35 in one pile.
9 Explain how Rick can use the pattern to find the number of cards in the other pile.
10 Describe what happens in addition patterns where the sum stays the same but
you change the numbers you add together.
Try It Another Way
Work with your group to use the tables to show patterns with addends and sums.
11 Fill in the missing numbers.
Addend Addend
20
30
12 Fill in the missing numbers.
Sum
Addend
Addend
Sum
100
100
100
100
100
5
8
10
16
20
20
25
25
25
25
25
80
65
60
50
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Part 3: Guided Practice
Lesson 7
Connect It
13 Explain: Izzy noticed a pattern in the addition table. She found a diagonal that
had all 5s in it. Fill in the table below on the right to show the addends.
0
1
2
3
4
5
0
0
1
2
3
4
5
1
1
2
3
4
5
6
2
2
3
4
5
6
7
3
3
4
5
6
7
8
4
4
5
6
7
8
9
5
5
6
7
8
9 10
Addend
0
Addend
4
3
2
4
5
Sum
5
5
5
5
5
5
Explain why the 5s form a diagonal line.
14 Examine: Jace counted to 50 by fives. Annabel counted to 50 by tens. What
numbers did both Jace and Annabel say? Explain why some of the numbers they said were the same.
15 Determine: Pat saw an odd number of birds on Monday and an even number of
birds on Tuesday. Is the total number of birds he saw odd or even? Explain how you know this, even though you don’t know how many birds he saw.
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Part 4: Common Core Performance Task
Lesson 7
Put It Together
16 Look at the multiplication table below.
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
3
0
6
9
12
4
0
4
12
20
28
36
5
0
5
15
25
35
45
6
0
6
12
18
24
7
0
7
14
21
28
8
0
9
0
18
15
18
21
36
35
18
27
48 54
49
40 48
9
24
63
64
36 45 54 63
81
AFill in the missing numbers.
BDescribe a pattern you see in the table.
CExplain why the pattern works the way it does.
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Focus on Math Concepts
Lesson 7
(Student Book pages 52–57)
Understand Patterns
Lesson Objectives
The Learning Progression
•Use hundreds charts, addition tables, and
multiplication tables to model addition or
multiplication patterns and explain why patterns
make sense.
This lesson builds on students’ previous informal
experiences with repeating words, shapes, or numbers.
In grade 3 students develop understanding of what a
pattern is and identify basic arithmetic patterns.
•Use (informally) number properties to find and
explain patterns.
A major focus in this lesson is exploring patterns in an
addition table. Students identify patterns in addends
that make the same sum. When comparing all the
facts, they notice that as the first addend increases
by 1, the second addend in each fact decreases.
•Use knowledge of even and odd numbers to find and
explain patterns.
Prerequisite Skills
•Use addition, subtraction, multiplication, and
division.
•Complete an addition table and a multiplication table.
•Recognize patterns in numbers.
•Know the difference between even and odd numbers.
Vocabulary
Students also explore patterns in a multiplication table.
A multiplication table has many patterns, so students
should have multiple experiences over time using the
table to find patterns. Some patterns help students find
products for unknown multiplication facts.
Students’ work with patterns in grade 3 provides a
foundation for describing, generating and analyzing
more complex patterns and eventually, examining
relationships between ordered pairs, coordinate graphs,
and studying proportional relationships and functions.
pattern: a series of numbers or shapes that follow a
rule to repeat or change
Teacher Toolbox
rule: a procedure that is followed to go from one
number or shape to the next in a pattern
Prerequisite
Skills
Review the following key terms.
Ready Lessons
even number: a number than can be divided into two
equal groups
Tools for Instruction
odd number: a number that cannot be divided into
two equal groups
Interactive Tutorials
Teacher-Toolbox.com
3.OA.D.9
✓
✓
CCSS Focus
3.OA.D.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using
properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be
decomposed into two equal addends.
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 2, 4, 6, 7 (see page A9 for full text)
L7: Understand Patterns
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Part 1: Introduction
Lesson 7
At a Glance
Students explore what a pattern is and how they can
describe patterns.
Focus on Math concepts
Lesson 7
A pattern is something that repeats. Sometimes we see patterns in shapes or letters.
Other times numbers make a pattern. Patterns can also be found when we use
numbers to add, subtract, multiply, or divide.
You can also see patterns in things around you. Look at the line of kids below.
think How can you describe a pattern?
You can also use numbers to describe this pattern.
1
60
2
3
4
5
6
7
8
9
The numbers 3, 6, 9, . . . tell where the girls are in line. Because the numbers in the
pattern do not repeat, you need to look for something that is done over and over to
get from one number to the next number.
52
•Explain that you can also attach numbers to the
sound pattern, like they did for the girls and boys
in line. Explore doing this with students.
What do you add to
get to the next
number in the
pattern? 3
Telling about what repeats in a pattern is called the rule.
The rule for the line of kids is boy, boy, girl.
•Read the Think question together. Explain that
there can be more than one way to describe a
pattern. If the students were numbered, then the
numbers 3, 6, 9, etc. show where the girls are in the
line. Emphasize that in order for shapes or numbers
to be in a pattern, something must be seen or done
to them over and over. Ask students for the next
3 numbers in the number pattern. [12, 15, 18] Have
them explain how they found these numbers.
•Explain to students that sounds can also be
patterns if they repeat over and over. Use a drum
stick or percussion instrument, such as a shaker
to make a sound pattern (ex: tap….taptap…tap).
Be sure to repeat the pattern at least 3 times so
students can hear how the sounds repeat. Direct
students to tap the same pattern on their desk.
Ask them to describe the pattern (one tap, two
taps, one tap and then it repeats).
3.oa.D.9
What are patterns?
•Introduce the Question at the top of the page. To
assess what students already know about patterns,
ask them to use their own words to explain what
patterns are and give examples. Use their ideas and
examples to clarify what patterns are and are not.
Direct students’ attention to the pattern shown on
the top of the page and have them explain what is
repeating.
Experience and describe a sound pattern.
ccss
Understand Patterns
Step By Step
Concept Extension
Part 1: introduction
L7: Understand Patterns
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Mathematical Discourse
•How can you prove that something is a pattern?
You must see the same numbers or shapes, hear
the same sounds, feel the same things over and
over.
•Describe this pattern: beep, beep, bop, beep, beep,
bop, beep, beep, bop. Can you attach numbers to this
sound pattern?
1, 1, 2, 1, 1, 2 or 2 beeps, 1 bop (2, 1, 2, 1)
•How do you know it’s a pattern?
Students should recognize that the sounds
repeat.
L7: Understand Patterns
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Part 1: Introduction
Lesson 7
At a Glance
Students explore ways to make number patterns visible
by using a grid.
Step By Step
•Read the Think question together as a class. Direct
students’ attention to the chart on the page. Explain
that the chart is not complete, but that the rule for
the pattern on the chart is “add 3.”
•Instruct students to work in pairs to study the chart
and discuss what they notice about the pattern that
could help them figure out what number the next
jump will show. Students may notice the difference
between each diagonal number is 9, so they know
that the next number will be 18 1 9, or 27.
•Have students read and reply to the Reflect directive.
You may ask students to work in groups to create the
pattern and share. This allows you to assess student
understanding and correct any misconceptions at
this point in the lesson.
•You may wish to give students practice circling
number patterns, instead of highlighting them. Point
out that when counting on a number chart, they may
wish to count out loud. This can be a good strategy
because some patterns may be easier for some people
to hear than to see.
STUDENT MISCONCEPTION ALERT: Some
students may see 2 white boxes between each shaded
box and describe the pattern incorrectly as a plus 2
pattern. Point out that the third box is shaded and
you can count each box starting with 1 to reach
number 3. Explain that you will have counted
3 boxes, so it is a plus 3 pattern.
Mathematical Discourse
•How could you prove that the chart shows a “plus 3”
pattern?
The third box is colored in for all the numbers
on the chart.
•Is there another way to describe the plus 3 pattern?
Go by three, count by 3, or multiples of 3. 2
white boxes, 1 shaded box, the multiples are in
diagonal lines.
L7: Understand Patterns
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Part 1: introduction
Lesson 7
think How do you know what numbers come next in a pattern?
You can use the rule to figure out what other numbers are
in the pattern. To get from one number to the next in the
pattern, you add 3.
The pattern is continued in the chart below.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22 23
Putting the numbers
in a chart helps me
notice things I might
not see if I just made a
list of numbers.
24 25 26
You may also notice other things about the pattern that can help you figure out what
numbers come next.
The numbers form diagonals in the hundreds chart. You can use them to tell the next
number in the pattern is 27.
The numbers in this pattern also alternate: even, odd, even, odd. Since 27 is odd, you
know that the number that comes after 27 will be even.
reflect
1 Write your own number pattern that has at least six numbers in it. Then, describe
the rule and one other thing you notice about the pattern.
Possible answer: 0, 5, 10, 15, 20, 25. the rule is add 5. the ones digits have
a pattern of 0, 5, 0, 5, 0, 5.
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Hands-On Activity
Create visual patterns on a number chart.
Materials: multiple copies of 1–100 number charts,
highlighters or crayons.
•Students benefit from many opportunities to
create and describe a variety of plus and minus
patterns on a number chart.
•Ask students to highlight or shade a plus 2
pattern. On another chart have them highlight a
plus 4 pattern. Give students opportunities to
describe and discuss the patterns for plus 2 and
plus 4. Then have them compare the two patterns.
Ask: How are the patterns alike? Different? Have
them highlight a plus 6 pattern and a plus 3
pattern on different charts. Ask them to describe
any patterns they see. Ask: How are the patterns
alike? Different? How do know it’s a pattern?
•You may wish to create and make copies of a
hundreds chart that begins with 100 and goes
to 1. Students can use the chart to show
subtraction patterns.
61
Part 2: Guided Instruction
Lesson 7
At a Glance
Part 2: guided instruction
Students explore a different way of thinking about
patterns. They see the pattern as a “doing” pattern in
which 10 is taken from one pile and given to another pile.
explore it
use the information below to help you think about patterns in addition.
Rick has a pack of 100 baseball cards and likes to sort them into 2 piles.
Step By Step
He notices that when he has a pile of 20 cards, the other pile has 80 cards.
When he has a pile of 30 cards, the other pile has 70. When he has a pile
•Work through the first four questions together.
Explain that each grid shows how Rick split up
the pack of 100 cards. On the board write
“20 1 80 5 100.” Ask students how he split the cards
up the next time. Underneath the first problem, write
“30 1 70 5 100.” Writing the addends on the board
gives students one more way to think about problem
and the pattern.
of 40 cards, the other pile has 60. Finally, when he has a pile of 50 cards,
the other pile has 50 too.
2 Rick shaded 20 squares in the first grid to show how many baseball cards were
in the first set of piles he made. Shade the rest of the grids to show the other
sets of piles.
3 What do the shaded squares show in each grid?
the number of cards in the first pile
•Tell students that they will have time to work
individually on the rest of the Explore It problems on
this page and then share their responses in groups.
•As students work individually, circulate among them.
This is an opportunity to assess student
understanding and address student misconceptions.
Ask questions such as: What is happening to the
shaded squares? As you compare all the grids, what is
happening that repeats? How do you know?
Lesson 7
4 What do the white squares show in each grid?
the number of cards in the second pile
5 What happens to the number of shaded squares as you move from one grid
to the next? the number of shaded squares gets bigger by ten.
6 What happens to the number of white squares as you move from one grid
to the next? the number of white squares gets smaller by ten.
7 Describe the rule for this pattern.
Possible answer: there are always 100 cards. every time you add some to
the first pile, you have to take the same amount away from the second pile.
54
L7: Understand Patterns
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•Take note of students who are still having difficulty
and wait to see if their understanding progresses
during the next part of the lesson.
•As a class, share the answers to problems 5–7.
Explain that a pattern can also be something that
you do or that happens with numbers over and over.
In this problem, as the number of shaded squares
gets bigger, the number of white squares gets smaller.
This idea may be new to some students. Point out
that getting bigger (shaded squares) and getting
smaller (white squares) are patterns.
Concept Extension
Help students see the pattern using number
sentences.
•Underneath the 30 1 70 on the board, ask
students for the number sentence to write when
Rick had a pile of 40 and 60. [40 1 60 5 100]
Complete the chart to show all the ways Rick
could have sorted his cards by groups of 10. Write
the number sentences for each way. Point out that
the two groups of cards always add up to 100.
62
Visual Model
•Some students may need additional support to
help them to see patterns in addends that equal
the same number. Use tiles or strips to show the
different ways to make 10. Then write the number
sentence for each way:
1 1 9 5 10
2 1 8 5 10
3 1 7 5 10
etc.
L7: Understand Patterns
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Part 2: Guided Instruction
Lesson 7
At a Glance
Students extend their thinking about addition patterns
where the sum stays the same, but the addends change
in a certain way.
Part 2: guided instruction
Lesson 7
talk about it
solve the problems below as a group.
8 The grid on the left shows Rick’s piles of 30 and 70. The grid on the right shows
Step By Step
Instruct students to continue to work in groups to
answer problems 8210. Walk around to each group,
listen to, and join in on discussions at different points.
Use the Mathematical Discourse questions to help
support or extend students’ thinking.
•Ask groups to share their thinking about
problems 8210. A possible answer for problem 9 is
that Rick always has 100 cards. If he starts with a pile
of 30, he knows the other pile must have 70 cards in
it. So, if he has a pile with 35 cards (5 more), then he
must have 5 less in the other pile (65).
•Direct the group’s attention to Try It Another Way.
Ask students to study the two tables and then ask a
volunteer to explain how they work. Instruct groups
to complete the tables.
•Invite two groups to draw the table on the board and
describe the patterns in the chart. Expect groups to
use precise math language in their explanations
using terms, such as addends, sum, rule, and pattern.
Students should share that all addends in the each
table equal the same sums and as one addend gets
larger, the other addend gets smaller.
SMP Tip: Students practice using precise
mathematical language when explaining their ideas
(SMP 6).
L7: Understand Patterns
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Rick’s piles of 40 and 60. Shade the grid in the middle to show what happens if
Rick put 35 in one pile.
9 Explain how Rick can use the pattern to find the number of cards in the other pile.
Possible answer: rick always has 100 cards. if he starts with the piles of 30
and 70, he has to add 5 to the 30 and take away 5 from the 70. this gets
him piles of 35 and 65.
10 Describe what happens in addition patterns where the sum stays the same but
you change the numbers you add together.
Possible answer: When you make one of the numbers you add bigger, you
must make the other number smaller by the same amount so that the sum
will stay the same.
try it another Way
Work with your group to use the tables to show patterns with addends and sums.
11 Fill in the missing numbers.
addend addend
20
30
sum
addend
addend
sum
100
100
100
100
100
5
8
10
16
20
20
17
15
9
5
25
25
25
25
25
80
70
35
40
65
60
50
50
12 Fill in the missing numbers.
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Mathematical Discourse
•How can using a chart help you to see patterns when
you add numbers with the same sum?
Students should recognize that charts often
allow you to see several steps of a pattern all
at once.
63
Part 3: Guided Practice
Lesson 7
At a Glance
Part 3: guided Practice
Students demonstrate their understanding of using an
addition table and charts to show addends with the
same sum. They reason about the patterns they find.
Lesson 7
connect it
13 explain: Izzy noticed a pattern in the addition table. She found a diagonal that
had all 5s in it. Fill in the table below on the right to show the addends.
Step By Step
•Discuss each Connect It problem as a class using the
discussion points outlined below.
0
1
2
3
4
5
0
0
1
2
3
4
5
1
1
2
3
4
5
6
2
2
3
4
5
6
7
3
3
4
5
6
7
8
4
4
5
6
7
8
9
5
5
6
7
8
9 10
addend
0
addend
1
2
3
4
3
2
4
5
1
0
5
sum
5
5
5
5
5
5
Explain why the 5s form a diagonal line.
Explain:
Possible answer: every time you make the first addend bigger, you have to
make the second one smaller. you move over and down the same number
•Focus students’ attention on the diagonal pattern of
fives on the addition table. Remind students that the
numbers along the top and sides of the table are the
addends and that the sums are found inside the table.
Be sure students understand how to use the table by
having them put one finger on the addend 3 on the
left and the addend 2 at the top and sliding their
fingers together to see the sum of 5.
of spaces and this makes a diagonal.
14 examine: Jace counted to 50 by fives. Annabel counted to 50 by tens. What
numbers did both Jace and Annabel say? 10, 20, 30, 40, 50
Explain why some of the numbers they said were the same.
Possible answer: some of the numbers are the same because there are
2 fives in a ten, so every other number will be said by both kids.
15 Determine: Pat saw an odd number of birds on Monday and an even number of
birds on Tuesday. Is the total number of birds he saw odd or even? odd
Explain how you know this, even though you don’t know how many birds he saw.
Possible answer. an even number is always made of pairs of two. an odd
•Have students work in pairs to complete the AddendAddend-Sum chart.
•Ask students to explain why the 5s form a diagonal
line. Expect students to say that the 5s are the sum
and as one addend on the left gets bigger, the other
added at the top gets smaller to make the sum of 5.
number is always made of pairs of two with one left over. When you add an
even and an odd, you have pairs of two with one left over. it doesn’t matter
which even number and which odd number you have.
56
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Examine:
•Ask two volunteers to come to the board and write
multiples of five and ten to 50. Have one of the
students circle the numbers they have in common.
Ask students how patterns and some numbers are
the same or similar in the two lists. Students should
see that it takes two fives to make ten, so they will
have some of the name numbers.
Determine:
•Students should recognize and understand that when
two even numbers are added, the sum is even and
when an even number and an odd number are added,
the sum is odd. You may wish to discuss why when
two odd numbers are added, the sum is even.
64
L7: Understand Patterns
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Part 4: Common Core Performance Task
Lesson 7
At a Glance
Part 4: common core Performance task
Students find patterns in a multiplication table and use
them to complete the table. They describe a pattern and
explain why the pattern works.
Lesson 7
Put it together
16 Look at the multiplication table below.
Step By Step
•Direct students to complete the Put It Together task
on their own.
•Let students know that they can circle a pattern on
the chart, but they need to describe the pattern in
words for letter B.
•As students work on their own, walk around to assess
their progress and understanding, to answer their
questions, and to give additional support, if needed.
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
3
0
9
12
4
0
4
8
5
0
5
10 15 20 25 30 35 40 45
6
0
6
12
18
24 30
7
0
7
14
21
28
8
0
9
0
3
8
9
6
12 16
10 12 14 16 18
15
18
20 24
24
27
28 32
21
36
36 42 48 54
35 42
49 56
63
16 24 32 40 48 56 64 72
18 27
36 45 54 63 72
81
a Fill in the missing numbers.
b Describe a pattern you see in the table.
Possible answer: all of the numbers in the 2 row are even.
•If time permits, have students share and describe
patterns they found.
c Explain why the pattern works the way it does.
Possible answer: all of the numbers are even because the row shows
Scoring Rubrics
groups of 2. 2 is even, and when you add even numbers, the sum is
always even.
A
Points Expectations
2
B
The student correctly fills in all the missing
numbers.
1
The student correctly fills in some of the
missing numbers.
0
The student does not fill in any missing
numbers or incorrectly fills in all the missing
numbers.
Points Expectations
2
The student finds a pattern and clearly
describes the characteristics of the pattern.
1
The student finds a pattern, but
characteristics of the pattern are not fully
described.
0
No pattern is described or what is described
is not a pattern and the description is
unclear.
L7: Understand Patterns
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L7: Understand Patterns
©Curriculum Associates, LLC
C
Copying is not permitted.
57
Points Expectations
2
The student shows understanding of the
pattern and explains clearly why it works
the way it does.
1
The student partially explains why the
pattern works the way it does.
0
The student is unable to explain why the
pattern works.
65
Differentiated Instruction
Lesson 7
Intervention Activity
On-Level Activity
Find patterns in a multiplication chart.
Further explore odd and even numbers using
addition tables.
Materials: blank multiplication chart
Asking students to construct a multiplication chart
can help them focus on and use patterns to complete
the chart and also help them learn multiplication
facts in the process. Ask students to fill in the first
row of the table. Remind them that the first column
also contains these same numbers and ask them to fill
in the column. Point to the 2nd row and remind
students that the numbers in this row are the
multiples or “count bys” of 2. Ask them to fill in this
row. Remind them that the column of twos contains
these same numbers and instruct them to fill in the
column. Have students fill in the fives and tens in the
same way. Then move on to the threes, then the fours.
Pause and direct students to look at the column and
row of 5s and ask what they see repeating. [0 and 5]
Ask whether each number is even or odd for the 5s.
Have them circle the repeating zeros. Have them
underline the repeating fives. Ask students to look for
other patterns they see. Instruct students to circle or
highlight several more patterns in the chart.
Materials: A blank chart with 11 rows and
11 columns or a completed addition chart with
addends to 10.
Have students highlight the addends along the side
and top of the table to make the sums easier to see.
Instruct students to work in pairs or a group and use
the addition table to help them explore questions,
such as: If you add 2 even addends, will the sum be even
or odd? If you add two odd numbers that are the same,
will the sum be even or odd? If you add any two odd
numbers on the chart, will the sum be even or odd? You
may wish to give each group or pair a different
question to explore. When pairs or groups share, ask
questions such as: Can you give some examples? Do you
agree with what said?
Do you think this is true for all whole numbers? How do
you know?
Challenge Activity
Materials: 1 inch (or larger) graph paper, scissors, tape, or use 2 colors of connecting cubes to make the train.
Ask students to work in pairs or small groups. Instruct each student to color in this pattern for every 4 squares in
one row on their graph paper: white square, white square, white square, colored square. Have the students cut
the rows apart and tape them together to make one long train. Be sure that the entire 4-square pattern repeats
throughout the train. The train should be at least 5 feet long.
Ask students to put a Star on the first square to show that it’s the front of the train. Then instruct students to cut
apart their train into squares of three starting the cutting from the front of the train. Have them cut at least 5
sections of 3 and stack them so the piece with the star is at the bottom of the stack. Have students notice and
describe the pattern that the colored squares make (diagonal going from left to top right).
Instruct students to put a star on the front of the remaining piece of train (the front square should be shaded) and
have students cut apart about 4 or 5 pieces of train that are 4 squares long. Ask them to stack the pieces with the
front of the train at the top.
Ask students to predict what the stacked squares will look like if they cut the remaining train into sections of 5
squares. Then have students put a star on the front square and cut apart the remaining train into rows of 5,
starting at the front of the train. Students can try this with cutting sections of 7 squares, and 8 squares. Have
student notice whether any of the patterns repeat in the stacked squares. Ask if they can use any of the patterns
to predict what trains of 10 and 11 would look like.
66
L7: Understand Patterns
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