Help your child visualize math concepts
For parents of students in the intermediate grades
The origins of mathematics lies in the concrete
and visual representations of numbers. Euclid’s
Elements, considered the first textbook of
mathematics, was written entirely from a
geometric viewpoint. Many of the words we still
use echo the visual representations of number.
For example, a square number such as 9, 16, or
25 is referred to as a square number since we
can arrange 25 objects into a perfect square (5
units by 5 units). Student struggles with math
can at times come from an over reliance on
procedures and “rules” (e.g. “two negatives
make a positive”). These rules are intended to
simplify the math but the end result is often that
students can’t recall which rule applies at what
times and very often they apply the rule in all
situations. Using concrete and visual
representations of the concepts can help
students reach a deeper conceptual
understanding that will allow them to make
connections as they move into more abstract
math in their later years.
Integers
Tools - Appropriate tools to help visualize
operations with integers include counters (two
colours) integers tiles or a number line.
Operations with Integers
The most important principle to remember when
representing operations for integers is the zero
principle.
+
-
Zero Principle: an equal number of positives
and negatives represent zero.
Strategies for ALL Operations
Whether students are working with whole
numbers, integers or fractions, try to use the
same understanding of the operations. We want
to avoid the situation where a student thinks that
any one of the operations is done differently
depending on the types of numbers.
Addition should be referred to as adding on to a
given quantity.
Subtraction should be referred to as either the
difference between two quantities or the removal
of a quantity from a given amount.
Multiplication should be referred to as multiple
groups of a given quantity.
Division can be referred to as sharing a given
amount among certain groups or creating
groups of a given amount from a set whole.
Begin by representing different integers in a
variety ways. Use combinations of negatives
and positives to reinforce the zero principle.
When adding, lay out the first quantity and then
add on the second quantity. Use the zero
principle to remove any zero pairs and
determine the remaining amount.
The key to subtracting is knowing that what is
being subtracted is in the given quantity (first
integer). Remove the amount represented by the
second integers if there is enough to remove. If
there isn't enough, introduce more integer chips
using zero pairs until you have enough that can
be removed.
Multiplication and division can be referred to in
much the same way as multiplication and
division with whole numbers. You have so many
groups of a given quantity or you are trying to
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determine how many groups can be made from
a certain quantity. The challenge is representing
problems that involve two negatives. This is
where it may become more abstract for
students. You can justify the result using
patterning but concrete models are difficult to
use to model these problems.
You may want to start with whole numbers
divided by a unit fraction before moving into
more complex problems. We want to model
division but asking how many parts of the
second fraction measure the first.
Fractions
Tools - Appropriate tools to help visualize
operations with fractions include fraction strips,
grids and counters (area model), Geoboards or
a number line. Try to avoid using circular
representations early on in student learning.
They become a challenge for students when
representing some fractions.
Operations with Fractions
Begin by representing equivalents fractions
using a variety of tools. Use the area model or
grids to divide a given fractions so you can
represent it using different equivalent fractions.
This will help students see that the value that
multiples by both the numerator and
denominator.
Using the area model to represent adding
should begin by creating a rectangle that has the
length dimensions of one denominator and the
width dimensions of the other denominator. This
will create parts of the whole that are the same
size (common denominator). Use the same
understanding of addition as students have for
whole numbers to add on the parts from the
second fraction to the first.
Model subtraction in the same way as addition
but this time we want to remove the parts from
the second fraction from the first to obtain our
result.
Multiplication can modeled in a variety of ways.
Do not forget that we should still refer to
multiplication in the same ways as multiplication
for whole numbers (e.g. I want half of one
quarter). Starting with unit fractions (fractions
with a numerator of one) helps students
visualize before moving to more complex
problems.
To model multiplication, you can overlay the
second fraction on top of the first to see the
overlapping parts of the whole that represents
the product. Or you can imagine two number
lines perpendicular to each other and model
each fraction on one of the number lines.
Division is more challenging but can be very
powerful when completed using the area model.
Tip sheet prepared by Paul Alves,
Marcia Charest, &
Jeremy Labrie, Teachers, PDSB
Workshop presented by Paul Alves
April 2, 2016
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