Math
Vocabulary & Concepts
algorithm
area
base-ten language
broken-line graph
chance
circle graph
circumference
composite number
conversions
fraction - decimal - percentage
units
coordinates
decimal
adding decimals
multiplying decimals
divisor
division
showing remainder results in two ways
estimating
expanded notion number
even number
factor number
factor chart
factor tree
flip
frequency diagrams
fraction
graphing
t-tables
weekly graph
conversions
height
intervals
irregular shapes
line plots
mass
mental math
multiple
multiplication
using arrays (or grids)
using expanded notation
decimal numbers
number patterns
number words
number sense
odd number
ordered numbers
ordered numbers
ordered pairs
patterns
percent
perimeter
place value
polygon
population
prime number
prime factor
probability
product
quadrant
quotient
ratio
reflection
rotation
rounding
side
slide
SI notation
standard form
tangram
translation
turn
volume
whole number
weight
word problems
Here are some
math glossaries online:
Harcourts
School
math.com
addition (regrouping): we regroup when we have enough in one place value
to move to another e.g. 7 ones plus 5 ones equals 12 ones or 1 tens and 2 ones.
We often need to regroup when we are adding numbers together..........see lesson
on traditional regrouping while adding; however, we also need to practice
adding numbers together without stacking them (e.g. 505 + 411 = 916). This
takes practice and understanding of the patterns in our number system, place
value.
algorithm a precise step-by-step procedure for solving a problem (our
goal is to limit our use of algorithms and help students try to create their
own understanding of the concept e.g seeing the patterns in the 9 times facts
rather than memorizing them)
area: the number of square units needed to cover a surface
e.g.
the area of a rectangle 2m by 6m takes 12 square meters to cover
see
using a factor chart to find the
possible lengths and widths for a given area
base-ten language: we use words to describe each
place value e.g. 104 one hundreds zero tens and four ones
broken-line graph: (also
see broken-line
graph)
chance: see probability
circle graph: a type of graph
using data showing how the parts are related to the whole
circumference: the distance
around a circle (also see circumference)
composite a whole number having more than two factors (math glossary)
(also see composites)
e.g. 4 has three factors: 1, 2, 4 (see example)
conversions:
fraction - decimal - percentage (also see converting visual )
- these number forms can all represent the same number
- for example 3/4 can be read as 3 ÷ 4
- using a calculator we find that 3 ÷ 4 = 0.75
- to find the % (the number out of 100), we need to
multiply the decimal by 100
- 0.75 as a percentage = 75 %
- therefore, 3/4 = 0.75 = 75 %
coordinates: see ordered pairs
decimal: a number with digits to the right of the decimal
point e.g. 5.48 (see adding or multiplying decimals). A decimal is another way
to write a fraction and we read decimals so that the fraction is heard. For
example, 5.48 is read "five and forty-eight hundredths). The 'and' tells
the listened that there is a decimal point. For a lesson on decimals see decimals.
dividend: the number that is to be divided e.g. 35 ÷ 5 = 7
35 is the dividend
division: division is the result of dividing
a large group (dividend) into smaller smaller groups (divisor); it is also the
inverse of multiplication
see:
division with a remainder and division with a
decimal answer
divisor: the number that divides the dividend e.g. 35 ÷ 5 = 7 5 is the divisor
expanded
notation: writing numbers out showing the place value of each digit e.g. 346 =
300 + 40 + 6
even: a number
divisible by 2 (the one's place value contains an even number e.g. 2, 4, 6, 8
or 0
estimating:
changing a exact number to an 'easy to use' number that is very close to the
exact number; we follow specific rules to do this. When rounding off a
number we look at the digit just before it. If that digit is a 5 or greater, we
increase the digit in our place value position by 1. If the digit is less than
5 we do not change it.
a) rounding off to a specific
place value e.g. rounding off 7 456 to the nearest hundreds = 7 500
b) front-end rounding means we
round off to the first (front) one or two digits e.g. 457 678 = 460 000
factor: a
number that divides evenly into another number (also see factors, example)
flip: see reflection
fraction: a part of a whole e.g. 1 out of 3 children
are playing hopscotch or 1/3 of the children
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3 squares are green (numerator)
6 squares in all (denominator)
so, 3 out of 6 squares are green, or 3/6
- you can reduce this to 1/2
example: for another view, go to Harcourt Math
Glossary
practice activity: go to www.rainforestmaths.com, choose your
grade level, choose Fractions; or go to http://nlvm.usu.edu/en/nav/vlibrary.html
tutorial: go to www.learnalberta.ca
(grade 5) for a couple of tutorials on fractions
frequency
diagrams: (also see frequency
diagrams)
height: the
distance from the bottom (base) to the top of a figure or solid shape
intervals:
(also see intervals)
irregular
shapes: shapes that do not contain symmetry (also see irregular)
line
plots: a 'picture' of data that shows the data along a number line (see line plot)
mass: see weight, a unit of weight
e.g. the mass of the pumpkin was 3 kg
mental math: when you use what you know to solve a question
in your head without the aid of a pencil and paper e.g. 7 X 8 =........ (4 X 8)
+ (3 X 8) = 32 + 24 = 56 (see strategies)
multiple: the set of numbers a specific number can go into evenly
e.g. 20, 30, 40 are multiple of 10 (see multiples)
multiplication: the product (total number of items) of the number
of equal groups e.g. we say 4 groups of 6 items = 24 items rather than 4
'times' 6 equals 24. The word 'group' means something whereas the word 'times'
does not see multiplying large numbers using arrays, using step-by-step and the 'traditional' method
number: a symbol or combination of
symbols representing a collection or quantity e.g. the number 15 represents a
specific collection of apples. We need to be able to see that this number can
be shown in many different ways e.g. 10 + 5, 3 X 5, 14 + 1, 30/2, (3 + 2) x 5,
20 - 5, (3 X 3) + (2 X 3)......
number patterns: an ordered set of numbers arranged
according to a rule (see Maths
Dictionary). For example a) 10, 8, 6, 4, 2, 0 (rule - 2), b) 10, 100, 1
000, 10 000, 100 000 (rule X 10)
number words: numbers written in word form e.g. ten (see Number Words for a printable page)
number sense: developing number sense is when the student
understands a concept or process by being able to recognize the pattern,
visualize the process, or recognize the concept in a new form as opposed to
memorizing a rule or algorithm they have been taught.
odd:
an odd number not divisible by 2 (one's place value contains an odd number e.g.
1, 3, 5, 7, 9
ordered numbers: number that appear in sequence either from
greatest to least or least to greatest (also see practicing ordering numbers)
ordered pairs: ordered pairs are
used to show where the x and y coordinates of a point are located on a plane
(also see ordered
pairs)
patterns: a number pattern is a constant or predictable
relationship between quantities. For example, 1 X 10 = 10, 2 X 10 = 20, 3 X 10
= 30 the rule for the 10s facts is that the input (e.g. 3) increases the output
previous output (2) by 10 each time .
percent: a number divided by 100
e.g. 40/100 = 40 % ; the ratio of a number to 100 e.g. 3/5 = 60/100
= 60%. Again, percent is another way to talk about a fraction, just like
decimals represent fractions
perimeter: the distance around a shape (see perimeter)
place value: the value of a digit depends on where it is
located, its place value e.g. 73629.57 the 6 is in the hundreds' place.
|
Millions |
Thousands |
Ones |
||||||
|
Hundred Millions |
Ten |
One Millions |
Hundred Thousands |
Ten Thousands |
One Thousands |
Hundreds |
Tens |
Ones |
Note: our
number system is based on the pattern: 'families' containing ones, tens,
hundreds. This makes is easier to predict the next section of numbers e.g.
Billions contains One Billions, Ten Billions, Hundred Billions
another way to
view this number to to identify each place value:
7 ten thousands
3 one thousands 6 hundreds
We can also
regroup the number so that 556 can be regrouped in a variety of ways. For
example:
5 hundreds 5
tens 6 ones = 556.....or
4 hundreds 15
tens 6 ones = 556.....or
2 hundreds 35
tens 6 ones = 556
Example: for a
more detailed view go to Harcourt Math
Glossary
Practice: go to
www.rainforestmaths.com, choose your
grade level > Numbers or go to http://nlvm.usu.edu/en/nav/vlibrary.html
Tutorial: go to
www.learnalberta.ca > log in as a guest
> Grade Five
polygon: a 2D shape formed by 3 or more lines e.g. rectangle, square,
triangle (see polygon
or geometric shapes)
population: the entire group considered in a survey e.g.
'all girls age 13 in the school' (also see population).
prime: a number divisible only by itself and 1 (see primes, example)
prime factor: a prime number that divides exactly into a
given number is one of its prime factors (example)
probability: the chance that an event will happen shown as a
ratio of the number of favourable outcomes over number of possible outcomes.
For some practice, go to www.rainforestmaths.com,
choose your grade and click on the 'Probability' section. (lesson)
product: the answer to a multiplication question (see product).
The product of 6 X 3 is 18.
quadrant: (see quadrant)
quotient: the answer to a division question. The quotient of 18 ÷ 3 is 6. (see quotient)
ratio: the comparison of two numbers by division 2:5 or 2/5 (2 out of 5
counters are red)
reflection: the flipped picture or mirror image of a shape
(see reflection)
rotation: the movement or the turning of a shape around a fixed point
(see rotation)
side: the edge
or line forming the border of a shape or object
slide: see translation
SI
notation: international system of units e.g. cm = centimeters, g = grams
standard
form: numbers written using digits e.g. 10
subtraction (regrouping): as with adding, we regroup or
borrow when we do not have enough in one place value to take away what we need
to ..... see lesson
subtraction (regrouping) on traditional regrouping while subtracting; however,
as with adding, we need to practice subtracting numbers without stacking them
(e.g. 525 - 411 = 114). Once again, this takes practice and understanding of
the patterns in our number system, place value.
tangram: a puzzle made of 7 polygon shapes
translation: the movement of a shape or figure along a
straight line
turn: see rotation
volume: the number of square cubes needed to fill a surface
e.g.
the volume of rectangular prism (length = 12m by width = 5m by height 2 m
takes 120 cubic meters to fill
whole number: the numbers (including 0) you would use to
count objects (therefore, whole numbers do not include fractions or decimals)
weight: mass, the mass is the amount in an object measured in e.g.
kilograms, pounds etc
word problems (or problem solving): Sometimes our
questions are in the form of a written word problem. To solve these, we often
follow five steps:
1) read the question twice, 2) draw a picture showing what
is happening in the problem, 3) write a number sentence show the problem, 4)
solve the number sentence, 5) write a clearly written sentence describing the
answer (remember to begin with a capital and end with a period). see example
It is also important to be able to create a word problem
when given a solution e.g. 5 + 14 =
large
numbers: here are some examples of larger numbers:
407 four
hundred six
3 589 three thousand eighty-nine
24 654 twenty-four thousand six hundred four
* remember that we do no use the word 'and' unless we are talking about a
decimal placement.
Here are some examples of decimal numbers. Remember that we
"read" the decimal as "and".
3.5 = three and five tenths
14.74 = fourteen and seventy-five hundredths
The 'How Tos'
The most
important part about adding and subtracting with decimals is to ensure that
your decimal points line up. You may add a zero to take a space if that helps:
|
set up: 3.500 |
e.g. 22.88 -
1.6 = set up:
|
* also lesson
Finding
Area: You use the following formula when find the area of a square or a
rectangle A = L x W
example: find the area of a rectangle with a length of 7 m and a width of 5 m
A = L x W
A = 7m x 5m
A = 36 sq m
Finding
Perimeter: You use the following formula when find the perimeter of a polygon.
P = (sum of all sides)
example: find the perimeter of a rectangle with a length of 7 m and a width of 5 m
P = (sum of all sides)
P = 7m + 5m + 7m + 5m
P = 24 m
Factors, Multiples,
Primes and Composites
1) method #1:
this is a factor chart. You create two columns of
numbers. On the left is a list of numbers you ask "does this number divide
into the number you are finding factors for. You start with '1' and work up
systematically. You stop when your next number also appears in the right-hand
column.
e.g. Find the
factors of 24
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Does this number divide into 24? |
Yes, how many times, or no. |
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1 |
24 |
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2 |
12 |
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3 |
8 |
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4 |
6 |
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5 |
no |
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6 |
4 |
Circle the
factors (pairs e.g. 4 & 6) and then write them in sequence: 1, 2, 3, 4, 6,
8, 12, 24
The prime
factors of 24 are 2 & 3 (by definition 1 is not a prime
number)
Finding Multiples:
Finding Multiples:
1) method #1: You will create a factoring tree
to find the prime factors for a number.
e.g. Find the prime factors of 24 using a factoring
tree. (for another example see math glossary)
24
/ \
2 12
/
/ \
2
2 6
/
/ / \
2
2 2
3
The prime factors of 24 are 2 and 3.
Since you cannot break the numbers down any smaller, these are the prime
factors of 24.
(Remember, 1 is not prime number.)
2) method #2: starting with dividing by 2, develop a list of factors
2 I24
2 I12
2 I6
3 2 X 2 X 2 X 3
= 24 so, the prime factors of 24 are 2 and 3
Finding Composites:
1) method #1: You create a chart to find whether a whole number has two or more
factors:
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Whole Number |
Factors |
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1 |
1 |
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2 |
1, 2 |
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3 |
1, 3 |
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4* |
1, 2, 4 |
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5 |
1, 5 |
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6* |
1, 2, 3, 6 |
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7 |
1, 7 |
4 & 6 are composite numbers, 1, 2, 3, 5, and 7 are not
1.
skip counting: 8 X 2 = 2, 4, 6, 8, 10, 12, 14,
16
2.
commutative property 9 X 2 = 2 X 9
3.
doubling: 2 X 8 = (8 + 8) = 16
4.
repeated doubling: 4 X 5 = (5 + 5) + 10 = 20
5.
distributive property: 4 X 15 = 4 X (10 + 5) = (4 X 10) + (4 X 5)
= 60
6.
halving: 12 ÷ 2 = 1/2 of 12 = 6
7.
repeated halving: 16 ÷ 4 = 1/2 of 16 =
8 1/2 of 8 =
4
8.
associative property (3 X 5) X 2 = 3 X (5 X 2)
9.
annexing then adding zero 62 X 10 = 62 X 1 and add a
0
a) Using
arrays or grid paper to break a question down into smaller parts.
- here is an easy one just to prove that it works
12 X 12 = 144
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10 X 10 = 100
2 X 10 = 20
10 X 2 = 20
2 X 2 = 4
144
The arrays help us break large numbers down into smaller,
easier to solve sections.
now, with a large number
e.g. fine the product for 12 X 36
grid paper 12 X 36
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This paper can be broken down into smaller groups
e.g.
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The purple, yellow, and pink sections are all
10 X 10 = 100 sections, so 3 X 100 = 300
The white section is 6 X 10
........................................................................ so 6
X 10 = 60
The brown, red, and dark green sections are all 2 X
10 = 20, so............... 3 X 20 = 60
Finally, the last little section is
..........................................................................2
X 6 = 12
If we add all of these products up, we solve the larger
equation.................................... 432
So, 12 X 36 = 432
b) Using expanded
notion to multiply large numbers
|
method #1 Now we can use expanded notation to help us
"see" the series of steps. We call this the 'step by step' method. |
method #2 This is the traditional method where we 'carry'
numbers to show our regrouping.
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Applying the process to problem
solving. To use with Power of Ten,
section 4, page 26
Building a Patio
You have 168 1 metre square patio blocks.
You must arrange into a rectangular shape.
Maximum length is 20 metres.
Use base-ten blocks or paper (e.g. graph
paper) to plan your patio.
In pairs, students create diverse ways to generate solutions. They share their
strategies, present, and support their suggestions. List strategies used e.g.
multiplication facts, division, scale diagrams.
solution
12 X 14 = 168
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Making Connections #1 to Factors and Prime Factors
Factor Chart
Factor Tree
168
168
1
168
/ \
2
84
2 84
3
56
/
/ \
4
42
2 2
42
5
no
/
/ / \
6
28
2 2
2 21
7
24
/
/
/ / \
8
21
2 2
2 3 7
9
no
10
no
11
no
12
14
13
no
14
Using
the factor tree, we find that the prime factors of 168 are 2 2
2 3 7 so 2 x 2 x 2 x 3 x 7 = 168
Other
factors can also be found using the 'cover-up' method.
-
cover one 2 and the remaining factors visible are2 x 2 x 3 x 7 which
= 84, thus 2 x 84 = 168
- try other 'cover-ups'
We can use a Factor Chart to find rectangles for
a given area.
For example, using the above chart. We know that length X width =
the area of a rectangle.
If the area is 168 square units, then all of our factor pairs (1, 168; 2,
84) provide the possible dimensions for this area. The area of a rectangle 6m
by 28m is 168 square meters. The following is a list of possible rectangular
shapes for the area of 168 square meters. (You may want to check it out using graph
paper.
1m by 168m
2m by 84m
3m
by 56m
4m by 42m
6m by 28m
7m by 24m
8m by 21m
12m by 14m
From this information, we know that 12m X 14m is the only rectangle that will
give us a shape that has no side longer than 20m. Thus, the answer to our patio
question, completed quickly and efficiently.
For more practice, go to graph
paper and scroll down to find a variety of square areas you can use. For
example, cut out the 25 square array and arrange them into as many rectangular
shapes as your can. Record these dimensions. Create a Factor Chart, or T-table,
and see how many you found.
Making Connections #2 to divisors
Use
the Factor Chart to list all of the divisors:
e.g. 2 x
84 168 ÷ 2 =
84
168 ÷ 84 = 2
or........
Use the 'cover-up' method to list all of the divisors:
e.g. 2
x 84 (2)x (2 x 2 x 3 x 7)
168÷ 2 = 84
168 ÷ 84 = 2
So,
12 X 14 = 168 is the only patio solution because any other rectangular shape
has a side of more than 20.
Other rectangular shapes that do not fit the criteria include
1 x 168
2 x 84
3 x 56
4 x 42
6 x 28
8 x 21
12 x 14 ............ the only possible solution
Multiplying Decimals
We solve multiplication questions the same, whether there are decimals or not. The only difference arises when we are finished the multiplication part. We then go back to the question, count the number of digits on the right side of a decimal and then count in that many digits in our answer.
1
12
X 36
72
360
4321
1.2 there are 2 digits to right of decimals
X 3.6
72
360
4.32 so, we need to count in two digits to place the decimal* see lesson on multiplying decimals
Dividing Large Numbers
a) Using arrays or grid paper to break a question down into
smaller parts. Using the original array, we divide 144 ÷ 12 = 12
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144 squares divided into strips of 12 makes 12 strips.
b) Division questions can
be written several ways.
When we have a remainder, we can show our answer in a
couple of ways. For example, we show the answer to the following question in
two ways: 147 ÷ 12
= a) 12 R3 , or b)
12.25 (we will round off to the nearest hundredths)
Look at the following examples:
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method #1 This is the traditional way we solve long division
questions. When we have a remainder, we write the answer in this way
12 R3 |
method #2 Although we solve long division questions the same
way as the traditional process, if we want to provide an answer in decimal
form when we have a remainder, then we need to keep dividing bringing down
more zeros, in these cases three zeros so that we can round our answer off to
the nearest hundredths. |
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Converting
from Fraction to Decimal to Percentage
Using
visuals helps students understand that these are different forms describing the
same number.
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This shape has ten equal parts. One part out of the ten parts is pink.
Therefore, 1/10 of the shape is pink.
Using decimals, we need to remember that the first place value after the
decimal point is the tenths place. One tenth of this picture is pink.
Therefore, we write 0.1 of this picture is pink.
Using percent, we want to know how many parts would be pink if this were a
shape made up of 100 parts. To do this, we need to multiply the fraction by
10/10 or 1 so that we know how many parts would be pink if there were 100 parts
in all. We need to keep the same ratio. 1/10 X 10/10 = 10/100 therefore, 10
squares need to pink. Ten out of the 100 parts are pink and, therefore, 10% of
the shape is pink.
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Converting the decimal to percentage means we multiply
the decimal by 100. So, 0.1 X 100 = 10.0, or 10 %. Later, we just move the
decimal over to the right two spots knowing that this is the result of multiply
a decimal by 100.
Our conversions look like this: 1/10 =
0.1 = 10%
Other examples, 1/4 = 0.25 = 25 %
1/2 = 0.50 = 50%
Graphing T-tables: Student
first fill in any missing data. They then take the data under each heading
(Input, Output) and graph it. The Input data is recorded along the bottom of
the graph while the Output data is recorded along the side of the graph.
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Weekly
Graphs:
teacher template
for overhead
student
template for working with graph data
Word Problems: To solve written word problems we often follow five steps: 1) read twice, 2) draw a helpful picture, 3) write a number
sentence, 4) solve the number sentence, 5) write a
clearly written sentence describing the answer
Example #1
Mike ran 5 miles one day and 7 miles the next day. How many miles did Mike run over
the two days?
day one:........5 miles
day two:........7 miles
5 + 7 = 12
Mike
ran
12 miles over the two days.
* Be sure that you answer the question in a full sentence using key words
in the question.
Example #2
A dog ran 1.25 km in a minute. At this pace, how far did he run in 12
minutes?
I.............I.............I.............I............I..............I.............I............I............I..............I............I...........I.............I
1.25 km
1.25
X.12
250
At
this pace, the dog ran 15 km in 12 minutes.
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