Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 70646
Multiplying by a Fraction Greater Than One
Students are asked to describe the size of a product of a fraction greater than one and a whole number without multiplying.
Subject(s): Mathematics
Grade Level(s): 5
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_MultiplyingByAFractionGreaterThanOne_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
Note: This task may be implemented individually or in small groups.
1. The teacher provides the student with the Multiplying by a Fraction Greater Than One worksheet and reads the following to the student:
Without finding the exact product, what can you tell me about the product of
and 2?
2. If needed, the teacher may prompt the student by asking, “Is it greater than ? Is it greater than two? Why or why not?
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to use what is known about the factors to correctly reason about the product.
Examples of Student Work at this Level
The student believes that when multiplying
by 2, the product will be smaller than
because multiplying by a fraction makes the product smaller.
The student insists that he or she must find the exact product before discussing it. The student determines the product incorrectly by multiplying both the numerator and
denominator by two and explains that the answer is
product is equivalent to
which simplifies to
, not greater than
, so the
. Or, the student determines the
page 1 of 3 product incorrectly as
or 2
or 2
and is unable to conclude that the product is greater than two and greater than
.
Questions Eliciting Thinking
If you multiplied seven by two, would the product be less than seven? Would it be less than two? Why or why not?
On a number line, where would
be placed in comparison to one?
Instructional Implications
Describe multiplication as a process of scaling. For example, if a 5 inch length is multiplied by two, it is scaled up to 10 inches or twice its original length. If a 5 inch length is
multiplied by
, it is scaled down to 2
inches or half of its original length. Provide opportunities for the student to explore multiplying a whole number length such as 5
inches by fractions that are close to one, such as
,
,
,
, and
,
,
,
. Guide the student to observe that when a whole number is multiplied by a fraction less than one, the product
is less than the whole number, and when a whole number is multiplied by a fraction greater than one, the product is greater than the whole number. Summarize the result
in terms of scaling (e.g., scaling up occurs when a number is multiplied by a factor greater than one and scaling down occurs when a number is multiplied by a factor less
than one). Provide expressions of the form r x n, where r is a scale factor and n is a whole number, such as
x 9 and
x 9.
Ask the student to determine if the product will be greater than nine or less than nine by reasoning about the size of the scale factor.
Explicitly model reasoning about the size of a product in relation to the size of a whole number factor. For example, when multiplying
be less than 53 since
is less than one.” Likewise, when multiplying 53 by a fraction such as x 53, say, “I know the product will
, say, “I know this product will be greater than 53 since is greater than
one.” Provide opportunities for the student to articulate this kind of reasoning.
Give the student additional opportunities to reason about the size of a product in relation to a whole number factor. Have the student use a number line or visual fraction
model to represent products of fractions and whole numbers. Guide the student to relate the size of the product to the size of the factor that is multiplying the whole
number.
Consider using the MFAS Task Multiplying by a Fraction Less Than One (5.NF.2.5) to assess if the student can relate the product of whole number and a fraction to the
factors.
Making Progress
Misconception/Error
The student cannot clearly explain the relationship between the size of the factors and the size of the product.
Examples of Student Work at this Level
The student knows that when multiplying
by 2 the product will be more than
and
more than two, but has difficulty explaining why, even with prompting.
The student is only able to provide a response about the product by finding the exact product. The student mentally adds
, or doubles it, and determines the product as
greater than two because
and
and concludes the product is
is equal to two.
Questions Eliciting Thinking
What whole number is
closest to? Is it greater than or less than one?
How do numbers change when we multiply by fractions?
What happens when we multiply by a fraction less than one? What about a fraction greater than one?
Instructional Implications
Guide the student to think about multiplying a whole number by a fraction as a process of scaling (e.g., scaling up occurs when a number is multiplied by a factor greater
than one and scaling down occurs when a number is multiplied by a factor less than one). Explicitly model reasoning about the size of a product in relation to the size of a
whole number factor. For example, when multiplying
fraction such as
x 53, say, “I know the product will be less than 53 since , say, “I know this product will be greater than 53 since a whole number, such as
x 9 and
is less than one.” Likewise, when multiplying 53 by a
is greater than one.” Provide expressions of the form r x n, where r is a scale factor and n is
x 9. Ask the student to determine if the product will be greater than nine or less than
page 2 of 3 nine by reasoning about the size of the scale factor and to explain his or her reasoning.
Allow the student to partner with another student at the same level and ask each to write products of the form of a fraction times a whole number. Then have the
students exchange cards and determine the size of the product in relationship to the whole number by reasoning about the size of the fraction.
Give the student a card with a fraction greater than one listed on it. Then call out a whole number to multiply by the fraction on the card. Ask the student to describe the
size of the product in relation to the whole number by reasoning about the size of the fraction. Repeat the process with different fraction cards to practice fractional
reasoning skills.
Provide an example of a calculator where the display only shows up to nine digits. Then ask the student to consider if the product of
and 999,999,999 can be displayed
on the calculator (adapted from the Illustrative Mathematics Project).
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
The student explains that
x 2 is greater than two because
is a little bit more than
one and that two groups of one are two, so the answer will be a little more than two. Additionally, the student explains that
x
2 is greater than two because multiplying a number times two will result in a product that is doubled. The student explains that
is more than one, so when it is doubled, the answer is more than two.
Questions Eliciting Thinking
Suppose you change
to
, how would your answer change? What can you tell me about the product of
and 2 without
finding the exact product?
What do you think happens to the product when you multiply a fraction less than one by another fraction less than one?
Instructional Implications
Challenge the student to reason about the size of products when multiplying a whole number by a decimal. Include examples with decimals less than one and greater than
one.
Introduce the student to multiplying two fractions, each less than one, and ask the student to reason about the product based on scaling. For example, when multiplying
by
, the product should be less than
since
is less than one. Likewise, when multiplying
by
, the product should be less than
since
is less than one.
Encourage the student to demonstrate this using an area model.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Multiplying by a Fraction Greater Than One worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.5.NF.2.5:
Description
Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without
performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given
number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a
given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle
of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
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