Grade 4, Module 6
Core Focus
• Introducing the comparison model and exploring the relationship between
multiplication and division
• Differentiate between multiplicative and additive/subtractive comparisons
• Solving word problems involving comparisons
• Exploring the concept of mixed numbers and addition of mixed numbers
Multiplication
• In Grades 2 and 3, multiplication has been viewed as equal groups (sets) or equal
rows (area). In comparison model multiplication, there are two different-sized
groups, and one group involves multiple copies of the other.
6.2
Using Tape Diagrams to Make Comparisons
Involving Multiplication
Grace cut two strips of material. The first strip is 2 yards long.
The second strip is three times as long as the first strip.
What is the length of the second strip?
How do you know?
What multiplication sentence could
you write to figure out the length?
Jamal drew this diagram.
First strip
I«ll call the first strip
of material A .
The length of the
second strip is 3 � A.
2 yards
?
In this lesson, students are introduced to tape diagrams to make multiplicative
comparisons between two numbers
• Phrases such as “times as many,” “times as long as,” and “times as heavy as,” signal
multiplicative comparison. Comparison problems can be solved more easily by
drawing a picture or a number line. When multiplicative comparison problems have
a known total, they may be solved by dividing.
in his collection as Lora. How many action
•
15
© ORIGO Education.
Lora
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5
• Listen for multiplicative
comparison claims in
advertising on television,
online, and in print. Talk about
the meaning of the statements
“times as many” “times fewer
than,” etc. “Our prices are
three times lower than the
leading brand!” means that
if the leading brand costs
$21.00, this brand costs $7.00.
• Practice multiplication and
division facts.
Second strip
David
Ideas for Home
5
3 × 5 = 15
5
15 ÷ 3 = 5
• When doubling or tripling
a recipe, use multiplicative
comparison language, “We
need two times as much rice
as the recipe calls for, how
much rice do we need?”
Glossary
A tape diagram models
the difference between
addition and multiplication
comparisons, or subtraction
and division comparisons,
by using two strips (paper
or drawn) to compare
two quantities.
• Subtractive comparison (finding the difference) is often confused with multiplicative
comparison. “Sarah has 3 apples and Jake has 4 apples; how many more apples
does Jake have?” By subtracting 4 − 3 = 1, it is evident that Jake has 1 more apple
than Sarah.
• “Jake has 4 times as many apples” demonstrates multiplicative comparison.
Jake has 4 times as many as Sarah, so 3 × 4 = 12. Jake has 12 apples in total.
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Grade 4, Module 6
Fractions
• Earlier, students worked with improper fractions. In this module, the concept of
mixed numbers is first introduced by rewriting whole numbers as common fractions
E.g. 3 = 15
because 15
= 55 + 55 + 55 = 1 + 1 + 1 = 3.
5
5
Exploring Whole Numbers and Common Fractions
6.7
Any whole number can be written as a common fraction.
What are some fractions that are equivalent to 3?
What helps you figure out they are equivalent?
+4
4
0
4
4
+4
4
1
8
4
+4
4
2
12
4
4
8
4 is 1 whole and 4 is 2 wholes
so 12
is
the
same
as
3 wholes.
4
You make a jump of 4
4 between
each whole number.
3
In this lesson, students explore the patterns and relationships between whole
numbers and common fractions that represent them.
• Students are encouraged to think about various ways that mixed numbers can
be composed and decomposed into whole numbers and common fractions,
as well as improper fractions. Understanding the concept of “What is the whole?”
is key to becoming flexible with representing fractional quantities beyond 1.
6.8
Ideas for Home
• Point out mixed numbers
in recipes, ask your child
to convert mixed numbers
to improper fractions. E.g. 2
8
is the same as 3 .
2
3
• Talk with your child about
mixed numbers and
encourage three ways to
explain them: in words, by
drawing pictures, and by
writing in numerals.
Glossary
A mixed number is a whole
number and a common
fraction added together and
written as a single number
without the addition symbol.
Introducing Mixed Numbers
Claire and Carter share 5 licorice sticks.
If they share the sticks equally, how much will each person have?
Fractions that are greater
than 1 are called improper
fractions, which can be
rewritten as mixed numbers.
They will need to break one
of the sticks to share it. Then
they will have two whole pieces
each, plus half a piece each.
A mixed number is a whole
number and a common fraction
added together and written as
a single number without the
addition symbol.
Imagine all the licorice sticks were
broken in half before sharing.
How would you write each share
as a fraction?
2+
Will each person still have the same
amount as before? How do you know?
1
2
2
•
This area model can be 54 or
1 41 . While called “improper,”
this type of fraction is
acceptable to write and use
in mathematics.
1
2
Mixed numbers can be made by joining together amounts in different ways.
3
2
5
is the same as 3 +
1
5
+
1
5
3
2
5
is the same as 2 + 1 +
2
5
In this lesson, students are introduced to mixed numbers.
© ORIGO Education.
• Adding mixed numbers can be illustrated using area models, but this module focuses
on using the number line. Overall, it is a more flexible model that can demonstrate
various composing and decomposing strategies to add mixed numbers.
•
In this lesson, students use different strategies to add mixed numbers.
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