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Money - counting dimes, nickels, and cents
This coin is called one cent or one penny.We write 1 c or 1¢.
This coin is called one dime.It is worth ten cents - 10c.
Here you see two dimes - 10 cents and 10 cents -and four pennies. Total 24 cents.
Example exercises
1. How much money? Write down the amount in cents.
a. c.
d. f.
2. Use real money, or draw gray circles with "10" for dimes and orange circles with "1" for pennies.
a. 62¢ c. 24¢
d. 77¢ f. 30¢
This coin is called one nickel.It is worth five cents, or 5¢
= 6¢
= 8¢
Whenever you have five or more individual cents (besides whole tens), use the nickel.
Here you see three dimes - worth 30 cents -one nickel (5¢) and three pennies. Total 38 cents.
3. How much money? Write down the amount in cents.
a.
c.
d.
f.
4. Draw one nickel more - how much money now?
a.
c.
d. f.
5. Use either real money, or draw gray circles with "10" for dimes, gray circles with "5" for nickels, and orange circles with "1" for pennies.
a. 25¢
c. 14¢
g. 61¢
i. 27¢
6. You have some, but what if you add some more? Use real money or draw pictures to help.
a.
10¢ + 10¢ =
11¢ + 10¢ =
13¢ + 10¢ =
15¢ + 10¢ =
16¢ + 10¢ =
c.
40¢ + 20¢ =
53¢ + 10¢ =
55¢ + 5¢ =
56¢ + 20¢ =
58¢ + 30¢ =
PRACTICE:
Fact families and basic addition facts
A fact family is a group of math facts using the same numbers. In the case of addition/subtraction, you use three numbers and get four facts. For example, you can form a fact family using the three numbers 10, 2, and 12: 10 + 2 = 12, 2 + 10 = 12, 12 − 10 = 2, and 12 − 2 = 10. You can use fact families to reinforce addition/subtraction connection, and to help children memorize basic addition facts.When studying basic addition facts with sums up to 18, you can choose one number to study and find all the possible sums that make up that number. For example, let the child find all two-number sums that make up 12, and memorize those at a time. See below an activity that helps a student do just that.
Studying sums associated with 12
Color with one color as many balls as the first number shows. Color the rest some other color.
Fact families with 12
10, 2, 12
10 + 2 = __ 2 + 10 = __
12 − 10 = __12 − 2 = __
__, __, 12
7 + __ = 12 __+ __ = __
__ − __ = ____ − __ = __
9, __, 12
9 + __ = 12 __+ __ = __
__ − __ = ____ − __ = __
__, __, 12
6 + __ = 12
12 − __ = __
__, __, 12
8 + __ = 12 __+ __ = __
__ − __ = ____ − __ = __
After finding the different sums associated with 12, and memorizing those, it is time to practice. The problems below are only showing types of exercises that a student might do; the teacher should provide more of each type of problems.
Example exercise types
1. Subtract. For each problem write a corresponding addition fact.
12 − 5 = ____ + __ = __
12 − 8 = ____ + __ = __
12 − 4 = ____ + __ = __
12 − 3 = ____ + __ = __
12 − 2 = ____ + __ = __
12 − 6 = ____ + __ = __
12 − 7 = ____ + __ = __
12 − 9 = ____ + __ = __
2. Find the missing addends and compare the problems. Write the corresponding problem where you only add one-digit numbers.
38 + __ = 428 + __ = 12
77 + __ = 82__ + __ = 12
37 + __ = 42
78 + __ = 82
32 + __ = 41__ + __ = __
55 + __ = 62
26 + __ = 32
89 + __ = 92
Studying sums associated with 13
Color with one color as many balls as the first number shows. Color the rest some other color.
Fact families with 13
10, 3, 13
10 + 3 = __ 3 + 10 = __
13 − 10 = __ 13 − 3 = __
__, __, 13
8 + __ = 13 __+ __ = __
__ − __ = __ __ − __ = __
9, __, 13
9 + __ = 13 __+ __ = __
__ − __ = __ __ − __ = __
__, __, 13
7 + __ = 13 __+ __ = __
13 − __ = __ __ − __ = __
Example problem types
1. Subtract and compare the problems.a.
13 − 7 = __23 − 7 = __33 − 7 = __43 − 7 = __
b.13 − 6 = __23 − 6 = __63 − 6 = __93 − 6 = __
What do you note? Discuss with your teacher. 2. Subtract. For each problem, write a corresponding problem where you subtract from 13.
23 − 9 = __13 − 9 = 4
33 − 4 = ____ − __ = __
73 − 7 = ____ − __ = __
53 − 8 = ____ − __ = __
3. Continue the patterns!
__ − 9 = 84__ − 9 = 74__ − 9 = 64__ − __ = ____ − __ = ____ − __ = ____ − __ = __
25 + __ = 3335 + __ = 4345 + __ = 53__ + __ = ____ + __ = ____ + __ = ____ + __ = __
Studying sums associated with 14
Color with one color as many balls as the first number shows. Color the rest some other color.
Fact families with 14
10, 4, 14
10 + __ = __ __ + 10 = __
__ − 10 = __ __ − 4 = __
__, __, 14
8 + __ = 14 __+ __ = __
__ − __ = __ __ − __ = __
9, __, 14
9 + __ = 14 __+ __ = __
__ − __ = __ __ − __ = __
__, __, 14
7 + __ = 14
14 − __ = __
Example problems
1. Find missing addends! Remember how a missing addend problem can be solved by subtraction? For each problem, write a subtraction problem using the same numbers so that the numbers in the boxes end up the same.
9 + = 14
14 − 9 =
6 + = 14
14 − 6 =
5 + = 14
14 − 5 =
7 + = 14
__ − __ =
3. Connect with a line the problems to the answer. What do you notice?
39 + __ = 4436 + __ = 4418 + __ = 2430 + __ = 3487 + __ = 9415 + __ = 2479 + __ = 8476 + __ = 8485 + __ = 9467 + __ = 7420 + __ = 2428 + __ = 34
796584
56 + __ = 6470 + __ = 7447 + __ = 5458 + __ = 6429 + __ = 3455 + __ = 6490 + __ = 9466 + __ = 7458 + __ = 6435 + __ = 4469 + __ = 7427 + __ = 34
PRACTICE:
Adding 2-digit numbers mentallyThere are several strategies that you can use when adding two 2-digit numbers mentally.
Strategy 1: Break the second addend in parts
45 + 27 = ? Break the 27 into tens and ones. First add 20, then add 7:45 + 27 =
45 + 20 +7 =
65 +7 = 72.
1. Add by breaking the second addend into tens and ones.
52 + 26 =
52 + 20+6 = +6 =
38 + 35 =38 + + =
+ =
25 + 57 =25 + + =
+ =
62 + 19 =62 + + =
+ =
2. Add mentally.
34 + 35 =47 + 12 =33 + 56 = 27 + 26 =
19 + 18 =13 + 17 =34 + 18 = 45 + 35 =
Strategy 2: Add the tens and ones separately
45 + 27 = ? Add tens. Add ones. Add the two sums:45 + 27 =
(40 + 20) + (5 + 7) =
60 + 12 = 72.
3. Add by adding tens and ones separately.
36 + 22 =
(30 + 20) + (6 + 2) =
+ =
72 + 18 =
(70 + 10) + (2 + 8) =
+ =
54 + 37 =
(50 + 30) + (4 + 7) =
+ =
24 + 55 =
(__ + __) + (_ + _) =
+ =
36 + 36 =
(__ + __) + (_ + _) =
+ =
42 + 68 =
(__ + __) + (_ + _) =
+ =
3. Add mentally, using either strategy - or one of your own!
36 + 38 =23 + 57 =27 + 41 =62 + 35 =38 + 49 =
22 + 36 =47 + 34 =66 + 37 =27 + 24 =72 + 19 =
Adding three numbers
The basic strategy is simply break the numbers in parts and add them little by little. You can add all the tens and all the ones separately, or add in whatever parts are easy to add. You can add the two last numbers first or do whatever works!
33 + 28 + 16 = (30+20+10) + 3+8+6 =
60 + 17 =
27 + 39 + 38 = (20+30+7+9) + 38 =
50 + 16 + 38 =66 + 30 + 8 =
96 + 8 = 104
You can also add first those ones that would form a ten.
15 + 27 + 45 = (5 + 5 + 10 + 40) + 27 =
60 + 27 =
18 + 52 + 16 =
(8 + 2 + 10 + 50) + 16 =
70 + 16 =
4. Add mentally.
22 + 43 + 38 = 37 + 18 + 15 = 22 + 43 + 38 = 24 + 13 + 25 = 54 + 16 + 22 =
28 + 18 + 36 = 51 + 15 + 16 = 63 + 12 + 17 = 17 + 60 + 20 = 50 + 31 + 19 =
Subtracting 2-digit numbers mentallyThere are several strategies or methods to subtract two 2-digit numbers mentally.
Strategy 1: Subtract in two parts
57 − 25 = ? Break the 25 into tens and ones. First subtract 20, then subtract 5.57 − 25 =
57 − 20 − 5 =
37 − 5 = 32.
This strategy is best whenthere would be no 'borrowing'if done in columns
1. Subtract mentally by breaking the second number into tens and ones.
89 − 26 =
89 − 20 − 6 = − 6 =
56 − 35 =
56 − − = =
75 − 51 =
75 − − = =
2. How can you compare these expressions without actually subtracting? Write < or >
60 − 28 60 − 25 90 − 25 90
70 − 24 70 − 36 97 − 32 90 − 32
Strategy 2: Add up to find the difference.
Subtraction gives you the difference between two numbers. To find the difference, you can start at the smaller number, and add up till you get to the bigger number.When adding up, first complete the ten, then add whole tens, then ones again.
84 − 37 = ?37
+ 3 = 40
40 + 40 = 80
80 + 4 = 84I added 3, 40, and 4; total 47. 84 − 37 = 47.
3. Add up to find the difference.
65 − 26 =+
+
+
26 30 60 65
56 − 28 =24 − 55 =
72 − 18 =82 − 46 =
74 − 55 =63 − 34 =
4. Find missing addends. The same method works here. Think: first add up to whole ten, then see how much more you need.
a.13 + __ = 3037 + __ = 7028 + __ = 9054 + __ = 80
d.36 + __ = 6036 + __ = 6465 + __ = 8065 + __ = 83
Strategy 3: Subtract an easy number that is close, then correct the answer.
This strategy works well if the number you're subtracting is a little less than a whole ten. You instead
subtract the whole tens, and then add back however much the "error" was.74 − 39 = ?
74 − 40 = 34
34 + 1 = 35.Subtract 40 since it's close to 39. You subtracted one too
much, so add one back
5. Subtract mentally using the above strategy.
34 − 18 =42 − 29 =76 − 59 =
65 − 27 =55 − 38 =94 − 48 =
6. Subtract mentally, using any of the methods - or one of your own!
66 − 38 =93 − 57 =27 − 41 =62 − 35 =88 − 49 =
55 − 46 =74 − 28 =48 − 13 =74 − 53 =91 − 59 =
Rounding to nearest ten
Is 223 nearer to 220 or 230?Is 247 nearer to 240 or 250?
Is 256 nearer to 250 or 260?Is 283 nearer to 280 or 290?
Is 298 nearer to 290 or 300?Is 305 nearer to 300 or 310?
The symbol ≈ is read "is approximately to". In the example above, 223 ≈ 220247 ≈ 250
256 ≈ 260283 ≈ 280
298 ≈ 300305 ≈ 310
This process is called rounding to the nearest ten. Actually, 305 is equally close to 300 and 310, but it has been agreed that numbers ending in 5 are 'rounded up' or rounded to the next ten.
When you are rounding to the nearest ten, the rules are: If a number ends in 1, 2, 3, or 4, then round down. If a number ends in 5, 6, 7, 8, or 9, then round up.
Examples (rounding down):671 ≈ 670, because 671 ends in 1 423 ≈ 420, because 423 ends in 3 544 ≈ 540, because 544 ends in 4 202 ≈ 200, because 202 ends in 2
When you round up, then tens digit increases by one. Examples:766 ≈ 770, because 766 ends in 6.
Tens digit (6) becomes 7.435 ≈ 440, because 435 ends in 5.
Tens digit (3) becomes 4.
705 ≈ 710, because 705 ends in 5. Tens digit (0) becomes 1.
296 ≈ 300 because 296 ends in 6. Tens digit (9) would become 10, which meanswe have 10 tens, one hundred. So the number
is rounded to the next whole hundred.
Try to 'visualize' or 'see' the number line in your mind when rounding to nearest ten.
Practice
1. Place the numbers on the number line and round them to the nearest whole ten.
a. b. c. d. e.554 ≈ ____628 ≈ ____551 ≈____
641 ≈ ____585 ≈____625 ≈____
595 ≈ ____592 ≈ ____636 ≈____
567 ≈ ____604 ≈ ____605 ≈____
612 ≈ ____616 ≈ ____597 ≈____
2. Round these numbers to the nearest ten.
a. b. c. d. e.78 ≈ ____64 ≈ ____86 ≈ ____
139 ≈ ____
98 ≈ ____105 ≈____233 ≈ ____887 ≈ ____
654 ≈ ____283 ≈ ____403 ≈ ____566 ≈ ____
347 ≈ ____816 ≈ ____46 ≈ ____
705 ≈ ____
288 ≈ ____497 ≈ ____908 ≈ ____202 ≈ ____
We can use rounding to estimate sums and differences.Estimate the sum 78 + 54.Since 78 ≈ 80 and 54 ≈ 50, 78 + 54 ≈ 80 + 50 = 130.Estimate the sum 423 + 69.Since 423 ≈ 420 and 69 ≈ 70, 423 + 69 ≈ 420 + 70 = 490.Estimate the difference 377 − 45.Since 377 ≈ 380 and 45 ≈ 50, 377 − 45 ≈ 380 − 50 = 330.
3. Estimate these sums and differences.
a. 148 + 43≈ 150 + 40 = 190.
b. 678 + 45≈ ___ + ___ = ___
c. 89 + 56≈ ___ + ___ =___
d. 344 + 34≈ ___ + ___ = ___
e. 237 + 52≈
f. 66 + 42≈
g. 23 + 98≈
h. 458 + 31≈
i. 178 − 43≈
j. 278 − 56 ≈
k. 873 − 98≈
l. 771 − 37≈
4. Solve the word problems using estimation. Use rounding to estimate the needed sums or differences.
a) In September Mom used the Internet for 53 hours, in October for 28 hours, in November for 19 hours, and in December for 35 hours. How many hours approximately did she use it total during those four months?
b) Susan bought 4 pens. The first pen cost 29 cents, the second one 32 cents, and the third one 44 cents. She bought two of the first kind. Estimate how much money did the pens cost total.
c) Jack, Ben, and Rick have been helping Grandpa to do yard work during the summer. Jack has earned 39 dollars, Ben has earned 27 dollars, and Rick has earned 122 dollars.
How much more approximately has Rick earned than Ben?How much more approximately has Rick earned than Jack?How much have they earned approximately all together?
d) You have $180. Estimate: do you have enough to buy a sofa for $118 and a table for $45?
How about if you buy a lamp for $29, four chairs $19 each, and a table for $65?How about if you buy a table for $57, a bed for $110 and a lamp for $15?
Regrouping in Subtraction
PRACTICE:
In this lesson I show how to teach regrouping (borrowing) in subtraction. We start out by teaching the actual regrouping separately, and then transfer that step-by-step into the subtraction algorithm.
First, teach the students about regrouping as explained in the box below.
As a first step, we study breaking a ten-pillar into ten little cubes. This is also called “regrouping”, because one ten “changes groups” from the tens group into the ones.
Breaka ten.
4 tens 5 ones 3 tens 15 onesFirst we have 45. We“break” one ten-pillarinto little cubes.
Now we have 3 tens and 15 ones. It is still 45, butwritten in a different way.
Here is another example. First we have 5 tens 3 ones. We “break” one ten-pillar into 10 little cubes. We end up with 4 tens 13 ones.
Breaka ten.
5 tens 3 ones 4 tens 13 ones
Example exercises follow. Here, students practice regrouping by itself, without sutbracting anything.
Break a ten into 10 ones. What do you get? You can draw ten-pillars and cubes to help.
Break
a ten.
a. 3 tens 0 ones 2 tens ___ ones
Break
a ten.
b. __tens __ones __tens __ones
Breaka ten.
c. __tens __ones __tens __ones
Break
a ten.
d. __tens __ones __tens __ones
Let's study subtraction. The pictures on the right illustrate 45 – 17.First, a ten is broken into 10 ones. So, 4 tens 5 ones becomes 3 tens 15 ones.After that, cross out (subtract) 1 ten 7 ones.
Breaka ten.
4 tens 5 ones 3 tens 15 onesCross out 1 ten 7 ones from the secondpicture. What is left? ___ tens ___ ones
The pictures on the right illustrate 52 – 39.First, a ten is broken into 10 ones. So, 5 tens 2 ones becomes 4 tens 12 ones.After that, cross out (subtract) 3 tens 9 ones.
Breaka ten.
5 tens 2 ones 4 tens 12 ones
Cross out 3 tens 9 ones from the secondpicture. What is left? ___ tens ___ ones
In the following exercises, the regrouping is done in the pictures. Students take note of that, then "cross out" or subtract something.
PRACTICE:
Breaka ten.
3 tens 6 ones 2 tens 16 onesa. Cross out 8 ones from the second
picture. What is left? ___ tens ___ ones
Breaka ten.
___ tens ___ ones ___ tens ___ ones
b. Cross out 2 tens 7 ones from the secondpicture. What is left? ___ tens ___ ones
In these exercises, students regroup, then subtract. The presentation uses "tens" and ones.
a. 5 tens 5 ones
– 1 ten 7 ones
4 tens 15 ones– 1 ten 7 ones
3 tens 8 ones
b. 7 tens 2 ones
– 3 tens 5 ones
__ tens __ ones– 3 tens 5 ones
__ tens __ ones
g. 8 tens 1 one
– 6 tens 5 ones
__ tens __ ones– 6 tens 5 ones
__ tens __ ones
h. 6 tens 3 ones
__ tens __ ones– 2 tens 8 ones
– 2 tens 8 ones __ tens __ ones
After this, the students should be ready to learn the usual form of subtracting where the numbers are in columns.
The picture illustrates subtracting 16 from 53. First, we break a ten into ten ones. Then we cross out 1 ten 6 ones.When the subtraction is written down in columns, we cross the “5” in the tens-column and write 4 above it. We also cross the “3” in the ones column and write 13 above it.This shows the same thing as the pictures: one of the tens is "broken down" into ten ones, so there is one less ten in the tens column, and 10 more ones in the ones column.Then we can subtract tens and ones separately.
5 tens, 3 ones 4 tens, 13 onesCross out 1 ten 6 ones from thesecondpicture.
What is left? ___ tens ___ ones4 13
5 3– 1 6
3 7
Now students can practice the normal way of subtraction in columns. The exercises below have the additional prompt for students to regroup before they subtract in columns.
a. 6 tens 3 ones → 5 tens 13 ones
Take away1 ten, 7 ones.
5 13
6 3– 1 7
c. 6 tens 0 ones → ___ tens ___ ones
Take away3 tens, 9 ones.
6 0
– 3 9
e. 3 tens, 5 ones → ___ tens ___ ones
Take away1 ten, 7 ones.
3 5
– 1 7
g. 7 tens, 6 ones → ___ tens ___ ones
Take away4 tens, 8 ones.
7 6
– 4 8
When students do normal subtraction problems, instruct them to always check their answer by adding.
Check: 4 16 1
a. 5 6– 2 7
2 9
2 9+ 2 7
5 6
Check:
b. 9 0– 2 8
+ 2 8
d. 9 0– 3 5
e. 8 2– 2 5
j. 5 5– 1 7
k. 3 1– 1 8
Multiplication and Addition
The symbol × indicates multiplication.Multiplication means to have many groups of the same size.
There are five groups, and each group has two elephants.
5 × 2 = 10
how many groups
how manyin each group
We can solve
it by adding.
5 × 2 = 2 + 2 + 2 + 2 + 2 = 10five times two elephants is ten elephants
There are three groups, and each group has four dogs.3 × 4 = 12
how many
groupshow many
in each groupWe can solveit by adding.
3 × 4 = 4 + 4 + 4 = 12three times four dogs is twelve dogs
1. Draw dots in groups to match the multiplications.
a. 2 × 6
b. 4 × 2
2. Fill in the missing parts. Write a multiplication. You can solve it by adding.
1. a.
___ groups, ___ scissors in each. ____ × ____ scissors = ____ scissors
___ + ___ + ___ + ___
b.
___ groups, ___ rams in each. ____ × ____ rams = ____ rams
___ + ___ + ___ + ___
c. d.
___ groups, ___ bears in each. ____ × ____ bears = ____ bears
___ + ___ + ___
___ groups, ___ carrots in each. ____ × ____ carrots = ____ carrots
___ + ___
3. Write an addition and a multiplication sentence for each picture.
a.
___ + ___ + ___ + ___ + ___ = ____ ____ × ____ = ____
b.
___ + ___ + ___ = ____
____ × ____ = ____
c.
___ + ___ + ___ = ____ ____ × ____ = ____
d.
___ + ___ = ____
____ × ____ = ____
4. Now it is your turn to draw. Draw balls or sticks. Write the multiplication sentence.
PRACTICE:
a. Draw 3 groups of seven sticks.
____ × ____ = ____
b. Draw 2 groups of eight balls.
c. Draw 4 groups of four balls. d. Draw 5 groups of two balls.
5. Draw a picture and solve the problems.
IIII IIII IIIIIIII IIII
a. 5 × 4 = ____
b. 4 × 6 = ____
c. 3 × 8 = ____
d. 2 × 10 = ____
Multiplying on a number line
Five jumps, each is two steps. 5 × 2 = 10.
Example problems
1. Write the multiplication sentence that the jumps on number line illustrate.
2. Continue and draw jumps to fit the multiplication problem.
6 × 4 =
5 × 5 =
3 × 10 =
Find 4 × 3 by skip-counting
Take four 'skips'. Add three each time. Notice where you land when 'skipping':
You landed at 3, 6, 9, 12. 4 × 3 = 12.
3. Add repeatedly (or skip-count) to multiply. You can use the number line to help.
3 × 2 =6 × 3 =4 × 5 =
3 × 10 =2 × 11 =3 × 7 =
4. Fill in the multiplication table of 2!
1 × 2 =2 × 2 =3 × 2 =
4 × 2 =5 × 2 =6 × 2 =
7 × 2 =8 × 2 =9 × 2 =
10 × 2 =11 × 2 =12 × 2 =
Multiplication in two ways (showing it is commutative)
PRACTICE:
Compare the two pictures:
4 4 + 4
12Three rows; four dogs in each row.
3 × 4 = 12
3 + 3 + 3 + 3 = 12Four columns; each column has three dogs.
4 × 3 = 12
Five rows; each row has two rams.___+___+___ +___+___
5 × 2 = ___
Two columns; five rams in each column.
___ + ___2 × 5 = ___
One row has five giraffes.___ giraffes
1 × 5 = 5
Five columns, each has one giraffe.___ + ___ + ___ + ___ + ___ giraffes.
5 × 1 = ___
You can do the same multiplication in two different ways but the result is the same. The order of numbers to be multiplied does not matter.
(Multiplication is commutative.)
1. Group the animals in two different ways and make the addition and multiplication facts to fit the pictures. In what case do you get the same fact either way?
a.
4 + 4___ × ___ = ___
2 + 2 + 2 + 2___ × ___ = ___
b.
_______________×___ = ___
_______________×___ = ___
c. d.
________________×___ = ___
_______________×___ = ___
________________ × ___ = ___
________________ ×___ = ___
2. Draw X marks and group them in two ways to illustrate the two ways to multiply.
a. 9 × 2 = ___ 2 × 9 = ___
b. 4 × 10 = ___ 10 × 4 = ___
Multiplying two ways on the number line
5 × 2 = 102 × 5 = 10
7 × 2 = 142 × 7 = 14
3. For each number line, write the two multiplication sentences that the arrows portray.
a.
b.
4. Write down the multiplication sentence. Then write the multiplication the other way, and draw the arrows for that multiplication sentence.
a.
b.
5. Skip-count to fill the multiplication table of 3. How does the picture connect with it?
1 × 3 =2 × 3 =3 × 3 =
4 × 3 =5 × 3 =6 × 3 =
7 × 3 =8 × 3 =9 × 3 =
10 × 3 =11 × 3 =12 × 3 =
How to memorize multiplication tables using a structured drill
This drilling is aimed at memorizing a certain times table. It should be used only after the child already understands the concept of multiplication.
When you are doing drills to memorize, explain to the child that the goal is to memorize the facts, to recall from memory, and not to get the answers by counting or some other method. Just like your child probably has memorized your address and phone number, now she/he is going to memorize some math facts. You can easily see if the student is trying to count because producing the answer takes much more time. You should expect the answers from the child immediately when you are drilling. If he/she doesn't know the answer by heart (from memory), then tell him/her the right answer.
Usually short drill sessions are best. You can drill for 5-10 minutes at a time, depending on the child.Try to have at least two sessions within a day though, as your schedule permits. Brain research shows us that forgetting happens fast, and that new information is retained far better if the first reviewing session is done within 4-6 hours of the first time learning. (This principle applies to anything new you are learning.)
Paper-pencil activities where the child is left alone, do not work really well for memorizing the facts - the child may get the answers by counting and not from memory. So it will take time from the teacher/parent. If you can, utilize older siblings in the drilling task too. Computers are great drillers since they won't get tired and you can usually choose a timed session where the child is then forced to produce the answers quickly. Children can actually enjoy the memorization process when they notice they are truly learning the facts and are able to go through the drills successfully. Computer programs and computer-based drilling can be very rewarding to children and let them enjoy memorizing times tables.
The method below has several steps from 1 to 5. You can work on only a few of the steps in one session, again, depending on the child's concentration and ability. Memorizing the table of 3 - in steps
Have a table to be worked on all ready written on paper. We will use here the table of three as an example.
1 x 3 = 32 x 3 = 63 x 3 = 94 x 3 = 125 x 3 = 156 x 3 = 187 x 3 = 218 x 3 = 249 x 3 = 2710 x 3 = 3011 x 3 = 3312 x 3 = 36
1. The first task is to memorize the list of answers, so to speak. Study first the skip-counting list up until the midpoint (3, 6, 9, 12, 15, 18). Have your child say it alound while pointing to the answers one by one with a finger or pen - thereby using many of his senses simultaneously. After he has gone through if a few times, ask him to repeat the list from memory. Try require the answers from your child, and not give them to her too easily, because ONLY by straining her mind will she make the effort to eventually memorize these facts. The mind is like muscles: it needs exercise to become stronger.
Require her to memorize this list both upwards and downwards. Continue this way until she can 'rattle off' the first list of 3, 6, 9, 12, 15, 18.With some tables, like table of 2, table of 5, or table of 10, point out the pattern in them. The pattern in table of 9 is more subtle but still usable.
2. Then tackle the last part of the list: 21, 24, 27, 30, 33, 36. Do the same things you did with the first part of the list.
3. Lastly, work with the whole list of answers. Practice the list UP AND DOWN until it goes smooth and easy.
This part may be enough for one day. But review it later in the day.
4. Next, practice individual problems randomly. You can ask orally ("What is 5 times 3?") or point to the problems on the paper, or use flashcards. However, I would recommend saying a question aloud and simultaneously pointing to the problem the child can see, because again, using multiple senses should help fix them in the mind better.
The goal at this stage is to associate each answer 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, with a certain multiplication fact (such as 7 x 3). You can also mix earlier tables that she already knows with these new problems, and drill both with flashcards.
5. The last step is to do this the other way round so that YOU say the answer, say 21, and the student has to produce the problem (from table of 3). Have the table handy, hide the problems, and point to the answers in random order.This one you can work the other way around: the student says answers, and you produce problems. Answer wrong sometimes, too, to check her out.
As an extension, you can say answers from several tables that you've studied, and the student gives the corresponding problem. Sometimes there are several answers: for example 36, 30, 24, and 20 are in several different times tables. This is an especially good exercise as it prepares to division concept and factoring.
The memorization won't probably happen overnight. On subsequent days, you can mix these drills 1-5 (and hopefully you don't need to concentrate on steps 1 and 2). This kind of drilling takes a little time and effort from the teacher, but it can be very effective. And, homeschoolers can obviously do some of it while going about other tasks, or while traveling in the car, etc.
While you are doing this table by table, you can also try to teach the process to your child, so that she will learn how to do memorization herself. She can hide the answers and try to produce the list in her mind. Other helpful ideas
Hang a poster with the 12x12 or 10x10 grid on the wall. Remind your child to glance at it a few times a day. It can work wonders for visual learners!
Hang beside it another, initially empty, poster, to which the child fills in those facts he has mastered. Recite the skip-counting lists or multiplication facts aloud just before going to bed. This can turn them
into mastered facts by the next morning.
The "easy way" to multiply in columns — the partial products algorithm
You can also do 5 × 34 vertically (in columns), using distributive property.
Multiply the ones first. Then multiply the tens and place
the result underneath. Remember,the 3 in 34 is signifying 30.
Then add.
34× 5
20
34× 5
20150
34× 5
20+ 150
1705 × 4 = 20 5 × 30 = 150
Multiply the ones. Multiply the tens. Add.
64× 8
32
64× 8
32480
64× 8
32+ 480
512
1. Multiply.
53× 2
61× 5
63× 5
35× 6
17× 5
99× 5
47× 8
35× 8
72× 5
77× 7
46× 6
88× 7
2. Write a mathematical sentence(s) for the problems. Multiply using the principle above (distributive property).
a) All seven houses on the Cat Lane have seven cats living in them. How many legs do those cats have total?
d) Jane bought seven cards and seven chocolate boxes for her seven friends. The cards cost total $14, and the boxes cost total $35. How much was her total bill? How much did one card cost? One chocolate box?
If you have a 3-digit number, the process is similar: 3 × 184
Multiply the ones first.
Then multiply the tens.
Then multiplythe hundreds.
Then add.
184× 3
12
184× 3
12240
184× 3
12 240300
184× 3
12240
+ 300
5523 × 4 = 12 3 × 80 = 240 3 × 100 = 300
Multiply ones: tens: hundreds: Add.
326× 3
18
326× 3
1860
326× 3
1860
900
326× 3
1860
+ 900
978
3. Multiply.353× 2
161× 5
327× 3
123× 3
103× 9
167× 5
199× 5
420× 2
245× 4
Can you figure out what was multiplied?
× 4
2 0+ 1 6 0
× 7
4 2+ 1 4 0
× 6
5 42 4 0
+ 6 0 0
Multiplying in columns - standard way
Often multiplication in columns is done like shown below. It is still based on the exact same principle (distributive property; you simply multiply ones and tens separately, and add. But this time the adding is done immediately after you multiply the tens.
Fill in missing numbers in these multiplications:
1 × 4
4 6 8
1 4 ×
8 7 0
3 9× 3
5
1 63
× 4
2
1 63
× 4
252
Multiply the ones first.4 × 3 = 12
Place 2 under the line at the ones place, but the tens digit (1) is written above the tens column as a little memory note. This is called carrying to tens.
Then multiply the tens, and add the 1 ten that was carried over.
4 × 6 + 1 = 25Total of 25 tens, which actually signifies 250. Write the 25 in front of the ones digit (2).
63× 4
12+ 240
252Compare to the method you learned in the last lesson.
In the calculation 4 × 6 + 1 = 25, the 6 and the 1 are actually tens. So in reality we calculate 4 × 60 + 10 = 250.
Look at other examples:
2 27
× 4
8
2 27
× 4
108
4 × 7 = 28 4 × 2 + 2 = 10
6 69
× 7
3
6 69
× 7
483
7 × 9 = 63 7 × 6 + 6 = 48
Compare the method from the last lesson with the one in this lesson:
75× 8
40+ 560
600
OR
4 75
× 8
0
4 75
× 8
600
5 × 8 = 40,4 is carried.
7 × 8 + 4 =56 + 4 = 60
You can choose which one you use. Discuss with your teacher.
Example problems
1. Multiply. Be careful with the carrying.
53× 8
51× 6
62× 2
46× 7
18× 5
39× 9
46× 2
17× 9
35× 9
With a 3-digit number you might have to carry twice, to tens and to hundreds.
3 238× 4
2
1 3 238× 4
52
1 3 238× 4
952
Multiply the ones first.
4 × 7 = 32Place 2 under the lineand carry the tens digit (3)to the tens' column.
Then multiply the tensdigit, and add the 3 tensthat were carried over.
4 × 3 + 3 = 15Place the 5 in tens place and carry the 1 into hundreds' column.
Then multiply the hundredsdigit, and add the 1 hundredthat was carried over.
4 × 2 + 1 = 9 Place the 9 in hundreds place.
Look at other examples. Compare to the method of the previous lesson.
2 127× 4
8
1 2 127× 4
08
1 2 127× 4
508
4 × 7 = 28 4 × 2 + 2 = 10 4 × 1 + 1 = 5
127× 4
2880
+ 400
508
2. Multiply.
123 151 462
× 8
× 6
× 2
106× 7
178× 5
109× 9
287× 3
367× 2
334× 2
157× 5
466× 2
104× 6
3. Word problems. Write a mathematical sentence for each one.
a) The school has 304 students. To go to the museum, they hired buses which can each seat 43 passengers. How many buses did they need?
b) The school also has 24 teachers. How many seats were left empty when all the students and all the teachers joined the trip?
f) Diaper service provided 90 diapers a week Was that enough for Anna and if not, how many more diapers did she need weekly?
Estimating products
Estimate first and use that to check the result. 8 × 67 = ?
67 can be rounded upto 70, or 67 ≈ 70. Therefore 8 × 67 ≈8 × 70 = 560.
5 67
× 8
356
5 67
× 8
536
Oops! 356 is far from the estimation 560.There must be an error. Can you find it?
This result sounds right,because it isfairly close to 560.
Before we start using estimation as a checking method, let's review the rounding rules.
When rounding to nearest ten, If the last digit is __, __, __, or __, round down to the previous ten. If the last digit is __, __, __, __, or __, round up to the next ten.
The symbol ≈ is used to indicate rounding, and is read as "is approximately" or "is about".Examples: 34 ≈ 30
598 ≈ 600 143 ≈ 140
255 ≈ 260 704 ≈ 700
705 ≈ 710
Example problems
1. First estimate the result by rounding the second factor. Then multiply to find out the exact result.
a.91 ≈ 904 × 91 ≈
4 × 90 = 360
91× 4
364
c.34 ≈ ____6 × 34 ≈
6 × ___ = ____
34× 6
e.99 ≈ ____5 × 99 ≈
5 × ___ = ____
99× 5
g. 48 ≈ ____ 48
3 × 48 ≈ 3 × ___ = ____
× 3
m.91 ≈ ____5 × 91 ≈
5 × ___ = ____
91× 5
2. These word problems illustrate some situations where you can use estimation.
a) 58 people are invited to the party; you are going to the store to buy supplies. You figure that for each person you need two cups, two plates, and three napkins. How many cups, plates, and napkins do you approximately need?
e) A can of beans costs 29 cents. A bag of lentils costs 42 cents. Estimate which is cheaper: to buy 8 cans of beans or to buy 5 bags of lentils.
h) Jill needs 21 inches of material to make a skirt. About how much should she buy for seven skirts?Now, add to the answer above 10 inches to make sure she has enough. The material costs 1 dollars per each 9 inches (or 4 dollars a yard). How much will it cost Jill to buy what she needs? Draw a picture! You can for example mark the 9-inch strips in your picture and use that to help.
i) What if you have a situation similar to the one in a) but just with 92 people. How many cups, plates, and napkins would you need? How many packages of each would you buy?
