Download Shining Term 2 Homework Answers

Hamilton Secondary Numeracy Project
Shining
Term 2
Homework Answers
Name ___________________________
Decimals and Fractions 1
1. Use a calculator to divide the numerator by the denominator to find the
decimal equivalents of 1/9, 1/99, 1/999… Can you predict the next decimal?
1/9
1÷9=
0.1111111
1/99
1 ÷ 99 =
0.0101010101
1/999
1 ÷ 999 =
0.001001001
1/9999
1 ÷ 9999 =
0.0001000100
1/99999
1 ÷ 99999 =
0.00001000010
1/999999
1 ÷ 999999 =
0.000001000001
2. Use a 0-9 dice or 1-9 cards (or remove the 10s, Jacks,
Queens, Kings and Jokers from a set of cards and use Aces as
1s, or write 1-9 on pieces of paper). Roll dice twice (if you roll
0 roll again) or choose two cards to make a fraction less than 1. Multiply it by
2
/3. For example, if you roll a 5 and a 2, multiply 2/5 by 2/3.
How many can you do in two minutes?
2 × 2 = 4
5
3
15
E.g. 1 x 2 = 2 = 1 E.g. 3 x 2 = 6 = 2
6 3 18 9
5 3 15 5
E.g. 4 x
7
E.g. 1 x 2
8 3
E.g. 8 x 2 = 16
9 3 27
E.g. 3 x 2 = 6 = 1
8 3 24 4
2= 8
3 21
=2=1
24 12
HSNP © Hamilton 2013
E.g. 2 x 2 = 4
7 3 21
E.g. 5 x 2 = 10
7 3 21
Shining Term 2
E.g. 1 x 2 = 2
9 3 27
E.g. 2 x 2 = 4 = 2
6 3 18 9
E.g. 5 x 2 = 10= 5
6 3 18 9
Page 1
Decimals and Fractions 1
If you have internet access, try this
Play Fractone at bit.ly/13xZBVl.
Choose ‘Pretty good’ or ‘I’m going for it’!
Click on pairs of fractions with a total of 1 as quickly as you can. Sometimes
you will need to click on pairs with different denominators, for example, 4/8
and 1/2. This is a computer game.
My time was
__________
just right
How did you find these problems?
too hard
too easy
Page 2
Addition 1
Remove the 10s, Jacks, Queens, Kings and Jokers
from a set of cards and use Aces as 1s (or use 1-9
cards or write 1-9 on pieces of paper).
Take two cards to make a fraction. Take two more to make another fraction.
Find the total of the two fractions using ‘smile and kiss’.
Only record the additions with a total of between 1 and 2. How many can you
do in five minutes with totals of between 1 and 2?
8+ 7
8
9
X
9 x 8 = 72
64/72 +
63/72 =
127/72
or 1 55/72
5+4
9 6
9 x 6 = 54
30/54 +
36/54 = 66/54
or 1 12/54
HSNP © Hamilton 2013
7+5
9 6
9 x 6 = 54
42/54 +
45/54 = 87/54
or 1 33/54
7+5
8 7
8 x 7 = 56
49/56 +
40/56 = 89/56
or 1 33/56
7+3
9 5
9 x 5 = 45
35/45 +
27/45 = 62/45
or 1 17/45
3+5
4 8
4 x 8 = 32
24/32 +
20/32 = 44/32
or 1 12/32
2+5
5 6
5 x 6 = 30
12/30 +
25/30 = 37/30
or 1 7/30
6+3
7 6
7 x 6 = 42
36/42 +
21/42 = 57/42
or 1 15/42
Shining Term 2
Page 3
Addition 1
If you have internet access, try this
Play Fruit Shoot Fraction Addition at bit.ly/11fmvhA.
Choose ‘Level 3’ and ‘Relaxed mode’.
Add the pair of fractions shown and click on the fruit with the answer.
Record the additions you complete below. What was your score out of 10?
This is a computer game.
Level 3a
Score =
just right
Level 3b
__
10
Score =
__
10
How did you find these problems?
too hard
too easy
Page 4
Subtraction 1
1. Roll a dice twice to make a fraction less than one.
Use ‘smile and kiss’ to subtract this fraction from 8/9.
Do three of these.
8
4
−
9X 6
9 x 6 = 54
- 36/54
= 12/54
or 6/27
48/54
E.g. 8/9 - 3/5
9 x 5 = 45
8 x 5 = 40
3 x 9 = 27
40/45 - 27/45 =
13/45
8/9 - 3/5 =
13/45
E.g. 8/9 - 2/5
9 x 5 = 45
8 x 5 = 40
2 x 9 = 18
40/45 - 18/45 =
22/40
8/9 - 2/5 =
22/40 or 11/20
E.g. 8/9 - 3/6
9 x 6 = 54
8 x 6 = 48
3 x 9 = 27
48/54 - 27/54 =
21/54
8/9 - 3/6 =
27/54 or 1/2
2. Solve the following by thinking about how many of each fraction are in
each number. For example, how many halves are in 20?
20 ÷ 1/2
6 ÷ 1/4
There are 2 halves in 1, so
there must be 20 times that
many in 20. 2 x 20 = 40.
10 ÷ 1/4 = 10 x 4/1 = 40
There are 4 quarters in 1, so there
must be 6 times that many in 6. 4 x
6 = 24. 6 ÷ ¼ = 6 x 4/1 = 24
3 ÷ 1/8
7 ÷ 1/6
There are 8 eighths in 1, so there
must be 3 times that many in 3. 8 x
3 = 24. 3 ÷ 1/8 = 3 x 8/1 = 24
There are 6 sixths in 1, so there
must be 7 times that many in 7. 7 x
6 = 42. 7 ÷ 1/6 = 7 x 6/1 = 42
30 ÷ 1/5
There are 5 fifths in 1, so there
must be 30 times that many in 30.
5 x 30 = 150.30 ÷ 1/5,30 x 5/1= 150
9 ÷ 1/10
There are 10 tenths in 1, so there
must be 9 times that many in 9. 10
x 9 = 90. 9 ÷ 1/10 = 9 x 10/1 = 90
HSNP © Hamilton 2013
Shining Term 2
Page 5
Subtraction 1
If you have internet access, try this
Play Adding and Subtracting Fractions Challenge at
bit.ly/13zR7P1.
Click to roll the dice. Draw a card. Add or subtract the
fractions. Carry on until you reach the finish.
My score was
__________
Play Fruit Shoot Fraction Subtractions at
bit.ly/14rZPwq. Choose ‘Level 3’ and ‘Relaxed mode’. Subtract the pair of
given fractions and click on the fruit with the answer. Record the subtractions
and answers below. What was your score? This is a computer game.
Level 3a
2/3
Level 3b
+ 2/6 = 1
__
10
just right
__
10
How did you find these problems?
too hard
too easy
Page 6
Multiplication 1
1. Find the squares of the numbers below. Then find the digital root of each
answer. For example, for 342 = 1156. Adding 1 + 1 + 5 + 6 gives 13. To get to a
single digit, add 1 and 3 to get the digital root of 4.
What do you notice about the digital roots of these numbers?
12
12 x 12 =
144
1+4+4
=9
9
23
23 x 23 =
529
5+2+9
= 16
1+6
=7
7
34
34 x 34
= 1156
1+1+5
+6 = 13
1+3=
4
4
45
45 x 45 =
2025
2+0+2
+5=9
56
56 x 56 =
3136
3+1+3
+ 6 = 13
1+3
=4
4
67
67 x 67 =
4489
4+4+8
+ 9 = 25
2+5
=7
7
78
78 x 78 =
6084
6+8+4
= 18
1+8
=9
9
9
I notice that______________All digital roots are 1, 4, 7 or 9___________.
Does the pattern work with 89?______It is 1___
Does it matter whether the larger digit is first or second?______No____
HSNP © Hamilton 2013
Shining Term 2
Page 7
Multiplication 1
If you have internet access, try this
Play the video about Alex’s Number Plumber at
bit.ly/12F8PnE.
Click on the picture under the video and enter the
same number as on the video.
Keep pressing ‘drop’ so that the last output becomes the next input. Click on
‘results table’ on the far right. What do you notice about the final digits of
each number? This is a computer game.
I noticed that…
Choose your own number to enter and see what happens. Keep re-entering
the output as the next input. Look at the results table.
Can you predict the pattern for a new number?
I predict the pattern to be…
just right
How did you find these problems?
too hard
too easy
Page 8
Division 1
1. Work to find the biggest four-digit number you can that is divisible by each
of its digits. Each digit must be different.
E.g. 1236 is divisible by 1, 2, 3 and 6, but you can do better than that!
1236
1236 ÷ 1 = 1236
1236 ÷ 2 = 618
1236 ÷ 3 = 412
1236 ÷ 6 = 206
1296
1296 ÷ 1 = 1296
1296 ÷ 2 = 648
1296 ÷ 9 = 144
1296 ÷ 6 = 216
1395
1395 ÷ 1 = 1395
1395 ÷ 3 = 465
1395 ÷ 9 = 155
1395 ÷ 5 = 279
3816
3816 ÷ 3 = 1272
3816 ÷ 8 = 477
3816 ÷ 1 = 3816
3816 ÷ 6 = 636
3195
3195 ÷ 3 = 1065
3195 ÷ 1 = 3195
3195 ÷ 9 = 355
3195 ÷ 5 = 639
HSNP © Hamilton 2013
9864
9864 ÷ 9 = 1096
9864 ÷ 8 = 1233
9864 ÷ 6 = 1644
9864 ÷ 4 = 2466
9864 is the biggest
possible, there are lots
of possibilities.
3612
3612 ÷ 3 = 1204
3612 ÷ 6 = 602
3612 ÷ 1 = 3612
3612 ÷ 2 = 1806
2196
2196 ÷ 2 =1098
2196 ÷ 1 = 2196
2196 ÷ 9 = 244
2196 ÷ 6 =366
4872
4872 ÷ 4 = 1218
4872 ÷ 8 = 609
4872 ÷ 7 = 696
4872 ÷ 2 = 2436
Shining Term 2
9648
9648 ÷ 9 = 1072
9648 ÷ 6 = 1608
9648 ÷ 4 = 2412
9648 ÷ 8 = 1206
9135
9135 ÷ 9 = 1015
9135 ÷ 1 = 9135
9135 ÷ 3 = 3045
9135 ÷ 5 = 1827
4236
4236 ÷ 4 = 1059
4236 ÷ 2 = 2118
4236 ÷ 3 = 1412
4236 ÷ 6 = 706
9162
9162 ÷ 9 = 1018
9162 ÷ 1 = 9162
9162 ÷ 6 = 1527
9162 ÷ 2 = 4581
1248
1248 ÷ 1 = 1248
1248 ÷ 2 = 624
1248 ÷ 4 = 312
1248 ÷ 8 = 156
Page 9
Division 1
2. Roll two dice (or choose numbers using 1-6 cards) to make
a fraction less than 1. Divide it by 1/4. For example, if you roll
a 5 and a 2, divide 2/5 by 1/4.
How many can you do in two minutes?
2 ÷ 1 = 8
5
4
5
3 ÷ 1 = 12
4 4
4
3 ÷ 1 = 12
5 4
5
1 ÷ 1 = 4
2 4
2
2 ÷ 1 = 8
3 4
3
4 ÷ 1 = 16
6 4
6
2 ÷ 1 = 8
6 4
6
3 ÷ 1 = 12
6 4
6
4 ÷ 1 = 16
5 4
5
1 ÷ 1 = 4
5 4
5
1 ÷ 1 = 4
4 4
4
2 ÷ 1 = 8
4 4
4
3. Two people are thinking of the same number less than 100. One divides it
by 3 and gets a remainder of 1; the other divides it by 20 and gets a
remainder of 3. What is the number? 43 ÷ 20 = 2 r3, 43 ÷ 3 = 14 r1
63 ÷ 20 = 3 r3
63 ÷ 3 = 21
just right
X
How did you find these problems?
too hard
too easy
Page 10
Decimals and Fractions 2
1. Use a calculator to find the decimal equivalents for 1/13, 2/13, 3/13 and so
on up to 12/13.
1/13 = 1 ÷ 13
0.076923
7/13 = 7 ÷ 13
0.53846153
2/13 = 2 ÷ 13
0.15384615
8/13 = 8 ÷ 13
0.61538461
3/13 = 3 ÷ 13
0.23076923
9/13 = 9 ÷ 13
0.69230769
4/13 = 4 ÷ 13
0.30769230
10/13 = 10 ÷ 13
0.76923076
5/13 = 5 ÷ 13
0.38461538
11/13 = 11 ÷ 13
0.84615384
6/13 = 6 ÷ 13
0.46153846
12/13 = 12 ÷ 13
0.92307692
Is there a pattern of recurring digits? Yes 076923 and 615384
Which fractions have the same pattern? 1,3,4,9,10,12 2,5,6,7,8,11
What do you notice about the sum of the digits? Both 27
Can you find any other interesting digit sums? The sum of the
fractions that share a pattern equal 39. i.e. 1 + 3 + 4 + 9 + 10 + 12 =
2 + 5 + 6 + 7 + 8 + 11 = 39. The fractions that share patterns are
pair bonds to 13. i.e. 1 + 12, 3 + 10, 4 + 9, 2 + 11, 5 + 8, 6 + 7.
HSNP © Hamilton 2013
Shining Term 2
Page 11
Decimals and Fractions 2
If you have internet access, try this
Play Fruit Shoot at bit.ly/12S4dHe.
Start with ‘Level 4’. Click on fruits with decimal equivalents to the given
fractions. Record your score.
Now have a go at ‘Level 5’! This is a computer game.
Level 4 score
Level 5 score
__________
__________
just right
How did you find these problems?
too hard
too easy
Page 12
Addition 2
1. Carry on Pascal's triangle so that you have at least 12 rows.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 11 55 165 330 462 462 330 165 55 11 1
Look at the second number in each row. Does this go into each number in the
row (apart from 1)? Look to see if there is pattern for this rule.
I noticed that… The rule is that if the second number in the row
is a prime number, it will go into each number in the row (apart
from 1).
HSNP © Hamilton 2013
Shining Term 2
Page 13
Addition 2
2. Use the internet to research Fibonacci sequences and spirals in nature.
Write about one fact which you found interesting.
I found out…
E.g. Leaves on a stem, fruitlets of a pineapple, flowering of artichoke….
3.
0
1
1
2
4
7
13
24
44
81
149
274
504
927
1705
3136
5768
10,609
19,513
35,890
66,012
121,415
223,317
410,744
755,476
1,389,537
This sequence is a twist on Fibonacci sequence. The fourth number is the sum
of the first three numbers, the fifth number is the sum of the previous three
numbers and so on. Continue the sequence so that you write at least 17
numbers.
What patterns can you find? Is there a pattern of odd and even numbers?
I noticed… that the sequence alternates with two odd numbers,
then two even numbers and so on.
just right
How did you find these problems?
too hard
too easy
Page 14
Subtraction 2
1. Josh says if you subtract a positive number from a
positive number, you will always get a positive answer.
What do you think? Explain your thinking with some
examples.
15 – 8 = 7
If the number subtracted is smaller than the other
number, it will be positive, if the number subtracted is
larger than the other number, it will be negative.
J
2. Roll a 0 to 9 dice (roll again if you roll a 0) and flip a coin to
determine whether the number is positive (heads) or negative
(tails). Repeat, then find the difference between the two
numbers. Draw a number line jotting if it helps. Record five subtractions.
4 4 – (-7) = 11
7
-7
0
4
E.g. -3 - 4 = -7
E.g. -7 - 2 = -9
E.g. -1 - (-2) = 1
E.g. 6 - 8 = -2
E.g. 5 - (-3) = 8
HSNP © Hamilton 2013
Shining Term 2
Page 15
Subtraction 2
If you have internet access, try this
Play Walk the Plank at bit.ly/16d1IQY.
Choose hair and skin colours for the person you
want to walk the plank. This is a computer game.
You will be asked a question. Roll the mouse over the pirates to see their
answers. Click the one you think is right. If correct, you’ll be asked to click on
the dice to move the person forward on the plank.
Click for the next question. Carry on until the game is complete.
Did you make the person
walk the plank? ________
Score __________
just right
How did you find these problems?
too hard
too easy
Page 16
Multiplication 2
Work out 1! 2! 3! up to 10! and record the answers below. Remember to use
your previous answer to help work out the next one.
1! 1
2! 2 × 1 = 2
Are all the answers odd or
even? All even (after 1!)
3! 3 × 2 × 1 = 6
Why? They all include x2
4! 4 × 3 × 2 × 1 = 24
5! 5 x 4 x 3 x 2 x 1 = 120
6! 6 x 5 x 4 x 3 x 2 x 1 = 720
Will all further factorials be the
7! 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
same? Yes
8! 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
9! 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 =
362,880
10! 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800
2. Now look at the digital roots.
For example, 9! 3 + 6 + 2 + 8 + 8 + 0 = 27. Add 2 + 7 to get 9, so the digital
root is 9.
What do the digital roots have in common? From 6! onwards
the digital root is 9.
What do you think will happen to further factorials? It will
always be 9. Why? The digital root of a number divisible by 9
is 9 and all further factorials will include x9.
What do you notice about the digital roots of multiples of 9? It
is always 9.
HSNP © Hamilton 2013
Shining Term 2
Page 17
Multiplication 2
3. Use a written method, to work out the following multiplications. Work out
the digital root for each.
Can you spot any patterns? ____Digital roots are 1, 4, 7 or 9_______________
12 x 21
Digital root
x
10
2
20
200
40
1
10
2
34 x 43
x
30
4
1200
160
3
90
12
56 x 65
1462 = 1 + 4
+ 6 + 2 = 13 = 4
20
3
30
600
90
2
40
6
6
60
3000
360
5
250
30
78 x 87
3640 = 3 + 6
+ 4 + 0 = 13 = 4
70
8
80
5600
640
7
490
56
just right
6786 = 6 + 7
+ 8 + 6 = 27 = 9
Digital root
x
40
5
50
2000
250
4
160
20
2430 = 2 + 4
+3=9
67 x 76
Digital root
x
60
7
70
4200
490
6
360
42
Digital root
x
736 = 7 + 3
+ 6 = 16 = 7
45 x 54
Digital root
50
Digital root
x
Digital root
40
x
252 = 2 + 5
+2=9
23 x 32
5092 = 5 + 0
+ 9 + 2 = 16 = 7
89 x 98
Digital root
x
80
9
90
7200
810
8
640
72
8722 = 8 + 7
+ 2 + 2 = 19 =
10 = 1
How did you find these problems?
too hard
too easy
Page 18
Division 2
The factors of 48 are:
1. If we add all the factors of
1 and 48
48 less than 48, we get 76.
2 and 24
1+2+3+4+6+8+12+16+24=76
3 and 16
48 is called an abundant
4 and 12
number because it is less than
6 and 8
the sum of its factors (without
itself).
32 has factors 1, 2, 4, 8 and
16 (apart from 32) and the
sum of these factors is 31, so
32 is not an abundant number
See if you can find some more abundant numbers!
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 X
1 + 2 + 3 + 4 + 6 + 8 + 12 = 36
36 > 24, so 24 is an abundant number.
The abundant numbers under 100 are:
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56
60, 66, 70, 72, 78, 80, 84, 88, 90 and 96
HSNP © Hamilton 2013
Shining Term 2
Page 19
Division 2
2. Use factor trees to find the prime factors of
five two-digit numbers. Try and make the
biggest tree that you can!
36
12
6
2
48
3
2
6
.
3
2
.
3 2
.
.
4
2
54
2
8
2
80
27
3
.
9
3
just right
4
2
3
.
96
20
2 2
4
10
2
2
5
.
24
2 4
6
2 2 2 3
How did you find these problems?
too hard
too easy
Page 20
Shining websites
Fractone
bit.ly/13xZBVl or
www.coolmath-games.com/0-fractone/index.html
Fruit Shoot Fraction Additions
bit.ly/11fmvhA or
www.sheppardsoftware.com/mathgames/fractions/FruitShootFractionsAddition.htm
Adding and Subtracting Fractions Challenge
bit.ly/13zR7P1 or
www.math-play.com/adding-and-subtracting-fractions-game.html
Fruit Shoot Fraction Subtractions
bit.ly/14rZPwq or
www.sheppardsoftware.com/mathgames/fractions/FruitShootFractionsSubtraction.
htm
Alex’s Number Plumber
bit.ly/12F8PnE or
nrich.maths.org/8387
Fruit Shoot
bit.ly/12S4dHe or
sheppardsoftware.com/mathgames/fractions/FractionsToDecimals.htm
Walk the Plank
bit.ly/16d1IQY or
www.math-play.com/integers-game.html
The links to the websites and the contents of the web pages associated with such links specified on this list
(hereafter collectively referred to as the ‘Links’) have been checked by Hamilton Trust and to the best of
Hamilton Trust’s knowledge, are correct and accurate at the time of publication. Notwithstanding the
foregoing or any other terms and conditions on the Hamilton Trust website, you acknowledge that
Hamilton Trust has no control over such Links and indeed, the owners of such Links may have removed
such Links, changed such Links and/or contents associated with such Links. Therefore, it is your sole
responsibility to verify any of the Links which you wish you use. Hamilton Trust excludes all responsibility
and liability for any loss or damage arising from the use of any Links.
Well done!
You’ve finished
Shining
Term 2