Download OA4-26 Patterns in the Times Tables

OA4-26 Patterns in the Times Tables
Pages 1–2
Goals
STANDARDS
4.OA.C.5
Students will identify multiples of 8 and multiples of 12.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can skip count by 8s and 12s
Can identify and extend increasing and decreasing sequences
created by adding or subtracting the same number
Can identify the core in a repeating pattern and extend it
column
even
multiple
odd
row
MATERIALS
BLM Patterns in the 2 Times Table (p. L-7)
BLM Patterns in the 5 Times Table (p. L-8)
Three dice for each pair of students
NOTE: Students who have trouble identifying multiples of 2 and 5 will
benefit from completing BLM Patterns in the 2 Times Table and BLM
Patterns in the 5 Times Table.
08
+1
+1
16
24
32
40
-2
-2
Introduce the multiples of 8. Write the first five multiples of 8 vertically.
Ask students if they notice any patterns. PROMPTS: What is the pattern in
the tens digits? (add 1 each time) What is the pattern in the ones digits?
(subtract 2 each time)
Have students write the next five multiples of 8 in a column and check
whether the pattern continues (it does: 48, 56, 64, 72, 80).
The pattern in the ones digit. Ask students to continue writing out all the
multiples of 8 that are less than 100 (88 and 96). Then ask students to cover
the tens digits and write only the ones digits in their notebooks. Write the
answer on the board:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6
ASK: What is the core in the pattern? (8, 6, 4, 2, 0) Remind students
what even and odd numbers are. ASK: Are there any odd numbers in the
pattern? (no; students can signal the answer using thumbs down) ASK: Can
a multiple of 8 be an odd number? (no) Why not? (because an odd number
has ones digit 1, 3, 5, 7, or 9, and multiples of 8 do not have such ones
digits) SAY: All multiples of 8 have 8 or 6 or 4 or 2 or 0 as the ones digit, so
they are even numbers.
Exercises
1. 273 is odd. Can it be a multiple of 8? (no)
2. Fill in the blanks to write each multiple of 8 as a multiple of 2.
Operations and Algebraic Thinking 4-26
L-1
a) 8 × 9 = (2 × ) × 9 b)
8 × 5 = (2 × ) × 5
= 2 × ( × 9) = 2 × ( × 5)
= 2 × = 2 × ACTIVITY 1
This game can be used
with the multiples of any
other number you would
like to reinforce.
“Eight-Boom” Game
Players count up from 1, each saying one number in turn. When a
player has to say a multiple of 8, she says “Boom!” instead. Example:
“1, 2, 3, 4, 5, 6, 7, Boom!, 9, ….” If a player makes a mistake, the other
players should correct her.
Advanced variation: When a number has 8 as one of its digits, the
player says “Bang!” instead. If the number is a multiple of 8 and has 8
as a digit, the player says both. Examples: “6, 7, Boom!, Bang!, 9,” or
“15, Boom!, 17, Bang!, 19.”
00
+1
+1
12
24
36
48
+2
+2
Introduce the multiples of 12. Use the same steps as for multiples of 8:
write the first five multiples of 12 vertically, then have students describe
how the tens digit changes as you move down the column, and how the
ones digit changes. Students should see that the ones digit of the multiples
increases by 2 and the tens digit increases by 1.
Multiples of 12 are even numbers. Again, have students write out the
ones digits of the first ten multiples of 12 and discuss the core of the
pattern. (0, 2, 4, 6, 8) ASK: Are there any odd numbers in the pattern? (no;
students can signal the answer using thumbs down) SAY: All multiples of
12 have 0 or 2 or 4 or 6 or 8 as the ones digit, so they are even numbers.
Exercise: Ron calculates 12 × 11 = 131. How can he tell he’s wrong? (131
is odd, so it can’t be a multiple of 12)
ACTIVITY 2
Materials: three dice for each pair of players
Instructions: Players each toss all three dice at the same time and
get a point if they can make a multiple of 12 using two outcomes and
multiplication. Players take turns tossing the dice and trying to score
a point. Each player should toss the dice five times, for a total of ten
turns per pair.
A player can choose one or two of the numbers rolled, but not all three
of them. A player can use either the number rolled or the difference
between the number rolled and 6. For example, if one of the numbers
rolled is 4, the player can use 4 or 2, since 6 − 4 = 2. If one of the
numbers is 6, the player can use 6 or 0, since 6 − 6 = 0.
L-2
Teacher’s Guide for AP Book 4.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Goal: To score at least five points as a pair by making multiples of 12.
Partners can give each other hints, but only when asked.
Examples:
Numbers
Rolled
Score Explanation
4, 3, 6
1
Choose 4 and 6 (4 × 6 = 24, a multiple of 12).
2, 5, 3
1
Choose 2 and 3 and multiply 3 by 6 − 2 = 4
to get 4 × 3 = 12.
1, 2, 4
0
The numbers that you can make from these
combinations are 2, 4, 8, 10, 16, and 20.
Have students explain why any combination containing 6 is a winning
combination. (Choose 6 and use 6 − 6 = 0. The product of 0 and any
number is 0, which is a multiple of 12.)
Extensions
1. “Who am I?”
a) I am a multiple of eight between 31 and 39.
b) I am the largest 2-digit multiple of eight.
c) I am the smallest 3-digit multiple of eight.
d)I am a 2-digit number larger than 45. I am a multiple of eight and
a multiple of five as well.
e) I am a multiple of twelve between 41 and 49.
f) I am the largest 2-digit multiple of twelve.
g) I am the smallest 3-digit multiple of twelve.
Answers: a) 32, b) 96, c) 104, d) 80, e) 48, f) 96, g) 108
Answers
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c)
9 15
8
4
3 6 27 0
16
21
12
18 33
24
20 28
30
36
39
40 44 32
45
48
42
60
56
51 54
52
57
Multiples of 3
Multiples of 4
d) in the central region
e) All multiples of 12 are
also multiples of 3 and
multiples of 4.
2. B
LM Patterns in the Times Tables (Advanced) (p. L-9) asks students
to sort multiples of 2 and 5 and then multiples of 2 and 8 into a Venn
diagram.
BLM Patterns in the 11 Times Table (p. L-10) reinforces the 11 times
table and shows students how to use patterns to multiply some 2-digit
numbers by 11, using a clever trick based on place value.
3. a) List all multiples of 3 from 0 to 60 (including 0 and 60).
b) List all multiples of 4 from 0 to 60 (including 0 and 60).
c)Use your lists in parts a) and b) to fill in a Venn diagram with the
categories Multiples of 3 and Multiples of 4.
d) Where in the Venn diagram are the multiples of 12?
e) Finish the sentence: All multiples of 12 are also …
Operations and Algebraic Thinking 4-26
L-3
OA4-27 Advanced Patterns
Pages 3–4
STANDARDS
4.OA.C.5
Vocabulary
decreasing sequence
difference
increasing sequence
sequence
term
---------------------------------- 8
---------------------------------- 4
---------------------------------- 2
---------------------------------- 1
Goals
Students will extend increasing and decreasing sequences using
addition, subtraction, and multiplication.
PRIOR KNOWLEDGE REQUIRED
Can identify and extend increasing and decreasing sequences
Can skip count by any number from 1 to 12
Can use a T-table to predict the terms of patterns
Patterns made by multiplication. Draw the tree shown in the margin on
the board and ask students to count the number of branches in each level.
Write the number of branches beside each level, as shown.
ASK: What happens to each branch? (it splits in two) SAY: This means
that the number of branches is doubled each time. So this sequence (the
number of branches at each level) was made by multiplication.
Ask students to find the sequence of differences (gaps) between the
number of branches at each level (1, 2, 4, 8, …). What do they notice? (The
sequence of the differences between the terms in the sequence is the same
as the sequence itself.)
Exercises: Write the first four terms of the pattern.
a) Start at 5 and multiply by 2 each time.
b) Start at 2 and multiply by 3 each time.
c) Start at 10 and multiply by 3 each time.
Comparing a pattern made by addition to a pattern made by
multiplication. Present two sequences:
SAY: The first sequence was made by addition. What number is added to
each term to get the next term? (10) The second sequence was made by
multiplication. What number do you multiply each term by to get the next
term? (2)
ASK: Which do you think will be larger, the 10th term of the first sequence or
the 10th term of the second sequence? Why do you think that will happen?
Accept all answers. Invite a volunteer to make a tally chart of the vote
results. Have students extend both sequences to the 10th term to check
the result. (The 10th term of the first sequence is 91 and the 10th term of the
second sequence is 512, so the latter is larger.)
A pattern made by adding and subtracting. Show another sequence:
3, 8, 6, 11, 9, 14, 12, 17, 15, …
L-4
Teacher’s Guide for AP Book 4.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Sequence 1: 1, 11, 21, 31, … and Sequence 2: 1, 2, 4, 8, …
3 6 9 12 15
8 11 14 17
To prompt students to describe and extend the pattern, you might write the
sequence in the form shown in the margin and ask students to describe the
patterns they see. Have students continue the sequence and explain the
rule by which it was made. Sample answers:
1.The top row is the sequence 3, 6, 9, 12, …. The rule is “Start at 3 and
add 3 each time.” The bottom row is the sequence 8, 11, 14, 17, ... and
the rule is “Start at 8 and add 3 each time.”
2.The general rule for the pattern (looking at all terms) is “Start at 3 and
alternately add 5 or subtract 2.”
Exercises: Write three more terms for each pattern.
a) 10, 12, 9, 11, 8, 10, 7
c) 2, 6, 18, 54
b) 2, 4, 8, 14, 22, 32
d) 98, 95, 90, 83, 74
ACTIVITY
Player 1 writes a sequence with at least three terms and Player 2 has
to find the next three terms. Player 1 checks the three new terms. For
each correct new term, the pair scores one point. If Player 2 is having
trouble determining the pattern, Player 2 can ask Player 1 for a hint
(but not a new term!). Hints could include “Look at the differences,”
or “I mixed two sequences,” or “I used multiplication.” For each hint
requested, the pair subtracts a point from their total. Players alternate
roles.
Discourage students from comparing their scores with those of
other pairs. Instead, students can use their scores to track their own
progress. Invite pairs to play repeatedly over several days and to look
for improvement.
Extensions
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9
4
8
5
6
7
3
2
1.Draw the number pyramids in the margin on the board. ASK: How can
you get the number in the top cell from the numbers in the bottom
cells? (the number at the top is the sum of the two numbers directly
below it) Draw a pyramid with numbers 2 and 3 in the bottom cells and
leave the top cell empty. ASK: What is the missing number? (5)
Then ask students to find the missing number in the pyramid in the
margin. (They will have to subtract 7 − 3 = 4.)
Have students work through BLM Number Pyramids (p. L-11). The
number pyramids on the BLM can be solved directly by addition and
subtraction. In larger pyramids with more than two rows, tell students to
look for a small three-square pyramid (two squares in a row with a third
square above them) in which any two squares are already filled.
Operations and Algebraic Thinking 4-27
L-5
15
4
5
After they complete the BLM, have students solve the pyramid with
three rows shown in margin using trial and error. Students should
guess the number in the middle square of the bottom row, then fill
in the squares in the second row, checking to see if they add up
to the number in the top square. Have students guess numbers
systematically, starting with 1. They can use a table like the one below
to organize the results and look for patterns.
The number in the middle of the bottom row
1
2
3
The numbers in the middle row
5, 6
6, 7
7, 8
The number at the top
11
13
15
Even with larger numbers, students will soon be able to see that the
numbers in the bottom row grow by 2 each time. This will allow them to
reach the correct number quickly.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
2.
BLM Problems and Puzzles (p. L-12) provides additional problemsolving practice.
L-6
Teacher’s Guide for AP Book 4.2