For older four year olds and 5 year olds.
skills to check or develop
Counting and Figuring from 1 to 20
A. Counting and Figuring 1 to 20
Activities
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Show students how to count aloud and then on paper from 1 to 20
aloud. Provide the word and one or two digit decimal representation
of each count or number. Repeat the exercises in items 1 to 7 (where
not too much nor too little) to assist in and test mastery of numbers
1 to 20.
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Physical and Pictorial Counts: Physically given a single bowls
with 0 to 20 objects (marbles, whatever) in each, find the total
number of objects by counting, label each bowl with the word and
digit form of the counts. Later, when the bowls are empty have
students fill each in accordance with their labels. Alternatively or
in parallel, students may do the foregoing with pictures. See the
More on Sequencing item for counting from 1 to 10.
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Comparision of Objects by Position: Identify or point to a place or position from
first to tenth? Where is the first, second, third, fourth, ...
nineteenth and twentieth element in a sequence. For example, students
may be shown a line of people walking and be asked to identify the
position of each in the line or queue.
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Comparision of Sets by Number: Given different set with 1 to 20 elements,
say which sets have the same number or different number of elements, which
has more than and which has less than another or several others.
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Counting Small Sets:
Show that different sets of 1 to 12 objects may have the same number.
[Counting how many is a property of sets, not their members]
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Picking Correct Number:
In counting, with examples, show how to select 2 to 20 objects from a
larger collection.
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Zero or None:
In counting, with examples, show how to say many objects there are
and when there are none, to say none or zero.
-
Connect or Join the Dots:
With examples, show how to arrange the numbers 1 to 20 in order, forwards
and backwards
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Activity: Join the Dots in sequence
puzzles:Â Â Parallel to learning the digits 1 to 9, their
meaning and order, students are most likely learning the or an
alphabet.  The simple task or puzzle in which students join dots
labeled with single digits or letters in proper sequence to arrive
at a geometric path, shape or picture tests and encourages the
ability to arrange whole numbers, single digit and larger, and
letters too in sequence. The geometric might represents a path
through a maze or a path take to buried treasure - anything that
will amuse. Letters too may come as small or capital, or in
various scripts.  The digits may be written not only using
arabic but also using roman numerals. Indeed tally marks could be
used in place of digits. Finally, the labels could be joined
forwards and backwards.Â
All sequencing recognition abilities and comprehension can be
tested by a joining the dot problem where the geometric path or
paths thus drawn outline a figure etc.. Â
Technical Note:Â Joining the dot exercises may provide a
context for this. Ask students to identify what comes before or
after a point in the sequence or path.Â
With this age group, continue to emphasize two counting practices or expectations when students
obtain different or incorrect results in counting
First Counting Practice A: The count (no
matter how large) should be independent of the counter. So when
counts differ or are incorrect, when two people obtain different
counts, a recount is required. Tell students to count twice or thrice
just to make sure their counts are correct.
Second Counting Practice B: Object may be
counted in different order, but in the end, the order in which
objects are counted does not affect the total count - no matter how
large. Show and confirm this observation by example.
As a teacher, older sibling or parent, you may deliberately make a
mistake in one to one tutoring to introduce this practice or practice
or principle.
B. Introducte Addition and Subtraction Physically
While the eventual aim to show students how do arithmetic with
decimals, the hands-on context and motivation for that comes from
physical addition and subtraction of groups of objects, as well as from
observing and counting the results.
-
Physical Addition: Physically given two bowls (sets) with 0 to
10 objects (marbles, whatever) in each, find the total number of
objects by counting - this sets the stage for addition of digits.
Drill and Practice is a must here. Choose examples so that
total number present is 20 or less.
Prequel to Counting Observation C: The order in which the
bowls are placed does not affect the total count.
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On paper, shown drawings of two set of dots (squares, circles, like
objects) with 10 or fewer than in each set, find the total number of
drawn dots (or like objects).
Technical Note: The regularity in this and the previous item
leads to the 10 by 100 addition table - explains why 3 + 4 = 7.
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Addition with Digits: Add pairs of numbers or counts verbally
or on paper. Fill in the 10 by 10 addition table to obtain and
record the results of all sums of pairs of numbers where each number
in the pair is from 0 to 9. Make sure physical addition is understood
before or while providing a written way to do
addition on paper.
Intermediate Steps: (1) Each single digit could be represented
along the side of the table by 0 to 9 dots or tally marks in
accordance with the value of the digit. Then the entry for say 3 + 4
would consist of 3 + 4 dots or tally marks or dots. The resulting
table could be generated without and then with the digits along side
the dots or tally marks along the sides and inside the table.
Finally, the tally marks could be dropped. (2) The foregoing could be
done with few digits, 1 to 4 or 1 to 7 etc, in the first instance.
Notation: Introduce column notation for addition of single
digits - that practice here helps sets the stage for place value or
column methods later.
8
+6
14
and the linear expressions 8 + 6 = 14 and 6 + 8 - 14 as well. Drill and
Practice is a must here. With result more than 10, students will not
be able to count on their fingers. But at this stage, learning to count
with objects actual or drawn, or with marks is part of a possible
development of reasoning skills.
Counting Observation C: The order of addition does not affect
the total count. Mention that to the students here and say that
explains why different counting by addition problems have the same
result.
Hands-On Viewpoint: The introduction to addition here is
based on the empirical observation that 7 units of something plus
another 5 units of the same material usually gives 12 units of that
material. Similar hands-on methods explain entries in the 10 by 10
addition table.
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Physical Subtraction: Take away or subtract a given number of
objects ( 1 to 9 say) from a larger set of objects (marbles, sweets,
pennies, and so on? Drill and Practice is a must here.
Technical Note: The regularity in this and the previous item
leads explains why 7 -3 = 4.
Intermediate step: To subtract 3 from 7 dots, show or indicate
that 7 = 3 + 4, and hence taking away 3 leaves a remainder of 4.
-
Subtraction with Digits: Subtract one digit number from second
one when the second one is larger - more than or equal to the first.
Notation 1 : Introduce linear expressions 9 - 5 = 4 as a
consequence of 9 = 4 + 5.
Notation 2: Introduce column notation for subtraction
9
- 5
and note the the result + the number subtracted, that 4 + 5 = 9, and
that should always be t he case. The foregoing introduces the
habit of checking a subtraction via an addition that needs to be
emphasized in future years. Drill and Practice sufficient to
develop and confirm skills is a must here
C. The Next Fraction - One quarter
-
The Fraction One Quarter: Show how to divide a pie, distance or
rectangle into four like parts, or four parts of equal value
Show students how to write the fraction in symbols and in words
- one quarter.
-
Recognition of Quarters and Halves Show how a quarter pie or
rectangle or square or distance is one of four equal or like parts? Show how half a pie or
rectangle or square or distance is one of two equal or like parts?
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Four Quarters Make a Whole, Show how a quarter pie or
rectangle or square or distance is made of four equal or like parts? Show how half a pie or
rectangle or square or distance can be given by two quarters.
-
Employ half and quarter hours, and half-days in talking about time.
D. Even and Odd Numbers upto at most 20
Counting in Groups of Two, Skip Counting, Even and Odd Whole Numbers
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Even Whole Numbers: Given the sequence 2, 4, 6, etc. predict
what number comes next? The answer could be 8, 10, 12.What pattern
does your charge see?
Note: Grade 1 students may continue the pattern and continue
with even numbers. As part of learning about this sequence,
students may asked to join dots labeled with elements of the
sequence. The continuation may be done by asking students to provide
the missing numbers.
The foregoing may be classified as skip counting by 2 instead of 1.
The sequence may also occur when counting in groups of two.
.. .. .. ..
.. ..
2
4 8 10
-
Odd Whole Numbers: Given the sequence 1, 3, 5, 7 predict or
guess what number comes next? The answer could be 9, 11, ... for a
sequence of odd numbers. Might be 11 for a sequence of primes. What
pattern does your charge see?
Note: Grade 1 students will not have studied primes. So they
may continue the pattern and continue with odd numbers. As part of
learning about this sequence, students may asked to join dots labeled
with elements of the sequence. The continuation may be done by asking
students to provide the missing numbers.
The foregoing may be classified as skip counting by 2 instead of 1. .
The sequence may also occur when counting in groups of two with one
left-over.
. .. .. ..
.. .. ..
1 3
5 7
9
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One half of Even and Odd Counts: Fractions in the first
instance may be understood in terms of geometric division of a pie, a
length, a rectangle, square or circle into two equal parts. Examples
of halves and further fractions may appear at home at meal time when
food is divided among family members. For numbers upto and including
20, show your learner how an even number of objects may be divided or
split into to groups of the same number. Then observe how division of
a odd group in to two equal subgroups leads to one left over.
-
Division by Dealing: Show how to divide upto 20 objects equally or nearly so
among two, three or more by handing out or dealing the objects, one at a time,
one after another in two, three or more separate piles. Observe when the
resulting piles are all equal in size/number.
E. Counting and Reading Measurements
-
Measure Integral Lengths: Count the number of inches
(centimeters) in lengths 1 to 20 inches (respectively centimeters) or
other units of measure where lengths are given by a whole number of
units.
Note: Students who understand the the fraction ½ may be
introduced to measure with half-units as well. 8½ cm for example. The
latter should be read aloud as eight and a half centimeters. Here 8 ½
serves as a multiplier - It describes how man centimeters are
in a length.
Show how to read or use a ruler and recognize when the zero mark of
the ruler is not an at an end. [Some mathematics exercise booklets
introduce this skil using drawings instead of rulers. Due to that
possibility, ruler use is explicitly mentioned.
-
Temperature Measures:
Show how to read a thermometers to find temperature.
Note: measures would be to the nearest whole unit, or nearest
half-unit. Here is a casual way to introduce negative numbers - when
the temperature is cold enough.
-
Comparison: Show how Compare objects by length, weight or
temperature: longer, shorter, lighter, heavier, cooler hotter,
Note: Lengths of strings can be compared without the aid of
measurement. Show students how to compare lengths without and with
the aid of measurement (rulers, tape measures) or units. p>
F. Geometry - Word Usage and Counting
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Naming/Identifying: With examples, show students squares,
rectangles, circles, triangles, and identify them by name. Then
test students on their ability to name these geometric shapes, and to
match figures with their names.
Note - a matter of taste: Triangles here should not all be
equilateral.
As parent or guardian, as you watch a child progress, there there may
be teachable moments that arise out of the blue, or may stem from
playful challenges - how far can you count? what shapes can you name?
What letters do you know? What is the shape of that door, window or
area? Older siblings and others may help and take pleasure in this.
The child may see learning more words, the names of objects, and
adjectives describing them or how many or how much, as part of
growing up and becoming stronger.
Show students the interior [insides] and perimeters of each figure. Point
to the exterior or outside.
Can he or she copy or reproduce them?
Can he or she point to the interior and perimeter of each.
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Length and Temperature Measures: Know how read or use a
ruler and thermometers? Can he or she estimate distance with a
ruler?
Note: measures would be to the nearest whole unit, or
nearest half-unit.
-
Show how to measure the perimeter of squares, triangles and
rectangles with the aid of a ruler, a string or the measure of each
side. Choose examples for which the perimeter will be 20 units or
less. Measurements may be based on counting how many inside of adding
lengths of sides.
-
Counting Sides: Instruct students to count the number of sides
in a triangle and rectangle. That should led to the knowledge that a
triangle has 3 sides and a rectangle has 4 sides.
Tell students about the interior and perimeters of each figure.
Can he or she copy or reproduce them?
Can he or she point to the interior and perimeter of each.
Can he or she measure the perimeter of squares, triangles and
rectangles
with the aid of a ruler, a string or the measure of each side.
Choose examples for which the perimeter will be 20 units or less.
-
Show how to use longer, shorter, larger, smaller, heavier, lighter,
yesterday, today, morning, afternoon, before, after, between in
counting and in measurement.
Note: For students who have started to read, work books may
give students pictures to interpret, and ask them to circle an answer
- one of the latter words. Or, teachers and tutors may give
activities to illustrate the uses of these terms.
G. Time and Date Matters
-
Telling Time and lengths of time (how long, duration)
Verify that students can read and recognize time on digital and
analogue clocks in terms of hours. Explain the meaning of waiting for
an hour, and learning how to compare lengths of time: Longer or
shorter.
At school and at home, activities - when and where and how long are
governed by the clock. Parents may set meal times, nap and
sleep-times, get-up times, play times by the clock. Parents and
teachers may tell student how long they must wait for an activity,
and will have or must do activity. The description may be terms of
the physical location of hands on a clock, or terms of hours and
fractions there-of, or in terms of minutes. With all that emphasis on
time, many students will know more than is demanded here in terms of
telling time. In societies without clocks, time may be told by the
position of the Sun, or be provided by church bells or local calls
for prayers.
Many homes activities involve time of day, day of week, day of the
month, month of the year. At ages 3 to 5, children may see that there
rise times, breakfast times, lunch times, supper times, wait times,
and bed times. At ages 3 to 5, children may also become aware of the
days of the week and the months of the year. When they count upto 31,
they may learn the day in the month. Birthdays and family festivities
expose children to the measurement of age, if not time, in terms of
years. Keeping track of height and mass (or weight) may provide a
practical introduction to measurement with a unit. Coins and money
may also help children develop their numerical or quantitative
skills.
-
Include in daily activities, their duration in 2 to 10 minutes,
more if that is understood. Talk about hours, half hours and even
quarter hours as well
In time and date matters, and in money matters, children may have many teachers at home as well
as in school because modern life is based on time described with words
and numbers.
H. Money Matters:
-
Show how to identify small coins (5 and 10 cent pieces) and how give
their values in terms of pennies. The handling of money in the home
and out povides an opportunities for this.
I. More Comparison Skills
-
Understand or explain the words: longer, shorter, larger, smaller,
heavier, lighter, yesterday, today, morning, afternoon, before,
after, between.
Note: For students who have started to read, grade
1 work books may give students pictures to interpret, and ask them to
circle an answer - one of the latter words. Or, teachers and tutors
may give activities to illustrate the uses of these terms.
-
Length and Temperature Comparison: Compare objects by length,
weight or temperature: longer, shorter, lighter,
heavier, cooler hotter,
Note: Lengths of strings can be
compared without the aid of measurement. With examples, show
students how to compare lengths without and with the aid of
measurement (rulers, tape measures), and imply the two different ways
are consistent (agree) in identifying what lengths are longer or
shorter, greater or smaller, more or less.
(Convex) Geometric Figures, their names and
properties:
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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