Grade 7
Number
and Operations
Standard 7-2: The student will demonstrate through the mathematical processes an
understanding of the representation of rational numbers, percentages, and
square roots of perfect squares; the application of ratios, rates, and
proportions to solve problems; accurate, efficient, and generalizable methods
for operations with integers; the multiplication and division of fractions and
decimals; and the inverse relationship between squaring and finding the square
roots of perfect squares.
The
indicators for this standard are grouped by the following major concepts:
The indicators that support each
of those major concepts and an explanation of the essential learning for each
major concept follows.
Indicators
7-2.1 Understand fractional percentages and percentages greater than one hundred.
7-2.10 Understand
the inverse relationship between squaring and finding the
square
roots of perfect squares.
7-2.2 Represent
the location of rational numbers and square roots of
perfect
squares on a number line.
7-2.3 Compare rational numbers, percentages, and
square roots of perfect squares by using the symbols ≤, ≥, <,
>, and =.
7-2.4 Understand the meaning of absolute value.
7-2.6 Translate between standard form and
exponential form.
7-2.7 Translate between standard form and
scientific notation.
In sixth grade, students were
introduced to whole number percents of one hundred or less. Seventh grade
students should extend this knowledge to percentages less than one and
percentages greater than one hundred. Using concrete models with enable the
students to connect the new learning to prior knowledge. Since students worked
with fractions that are less than or greater than one in third grade, they
should now be given opportunities to line that to fractional percents and
percents greater than 100%. Students should be given opportunities to develop
models using materials such as base-ten blocks, 10 x 10 grid paper, or graph
paper to help students visualize the connection of fractions to percents less
than one percent and mixed number to percents greater than 100%.
In third grade, students had
their first experiences with perfect squares as they learned basic
multiplication facts such as 4 x 4, 7 x 7, etc. In fifth grade, students were
exposed to the concept of squares when they determined the area of geometric
squares. When the terms squares and square roots are introduced, it is
essential that the connection is made between the squared number and the
corresponding geometric square. In other words, students should understand that
“find the length of a side of a square with area equal to 25 units” and “find
the square root of 25” are basically the same question. Students have been
exposed to inverse relationships for addition and subtraction in first grade
and multiplication and division in third grade. In seventh grade, the concept
of inverse relationships is expanded to include squaring and finding square
roots of perfect squares. Learning opportunities should include both models and
numbers.
In fifth
grade students compared whole numbers, decimals, and fractions. In sixth grade
the comparisons were continued and whole number percents were included. Now in
seventh grade the expectation is to locate rational numbers and square roots of
perfect squares on a number line. Students should explore the location of
fractions, decimals, percents, and square roots of perfect squares on a number
line. It is equally important that students justify the placement of these
representations on a number line, as well as understand the relationship to the
numbers between which a given value lies. Being able to justify the placement
on a number line will enable students to compare and order rational numbers,
percentages, and square roots of perfect squares using the symbols ≤,
≥, <, >, and =.
New to seventh grade students
will be an understanding of the meaning of absolute value. Student instruction
should focus on the fact that the absolute value of a number is the distance of
the number from zero. The absolute value of any number except zero is a
positive value. Students often have the misconception that distance can be a
negative number when determining absolute value of negative numbers. Students
need to be reminded that absolute value provides a distance and not a direction.
An understanding that distance is always a positive value is essential to
develop a solid understanding.
In sixth grade, students applied
strategies and procedures to determine values of powers of 10 up to 106.
In seventh grade, students build on that knowledge by translating between
standard form to exponential form and to scientific notation. In sixth grade, students represented whole
numbers in exponential form. Seventh grade is the first time students transfer
numbers between standard form and exponential form and scientific notation.
Students need to understand that in scientific notation the first number should
be greater than or equal to one and less than ten. Students need to work with a variety of numbers,
both very large and very small, as well as decimal and whole numbers.
Indicators
7-2.8 Generate strategies to add, subtract,
multiply, and divide integers.
7-2.9 Apply an algorithm to multiply and divide
fractions and decimals.
7-2.5 Apply ratios, rates, and proportions to
discounts, taxes, tips, interest,
unit
costs, and similar shapes.
In
sixth grade, students developed a conceptual understanding of an integer.
Seventh grade students generate strategies to add, subtract, multiply and
divide integers. Students should not
be expected to perform symbolic operations with integers. Spending the time to
fully explore strategies to perform integer operations will pay off in future
mathematics courses for students. Students should work with concrete models and
pictorial representations to build the foundation needed for eighth grade when
abstract/symbolic integer operations are performed. To support and promote
conceptual understanding of operations with integers, manipulatives should be
used and students should be allowed to generate algorithms for addition,
subtraction, multiplication and division before introduction to traditional
algorithms in eighth grade.
In sixth
grade, students generated strategies to build conceptual understanding of
multiplying and dividing fractions and decimals. Seventh grade is the first
time students are required to multiply and divide fractions and decimals symbolically
(numerals only). As a result, students should be given opportunities to relate
their prior concrete and pictorial experiences to the new symbolic operations. In
addition to building on those previous experiences, students should estimate
the products and quotients of problems involving fractions and decimals and use
those estimations as the basis for explaining the reasonableness of results
after actually solving. Furthermore, students should be given opportunities to
apply multiplication and division of fractions and decimals in context – not
merely perform the operations for the sake of multiplying or dividing. Teachers
should be alert to the misconception that all multiplication results in “a
bigger number”. Multiplication with fractions and decimals may result in
a smaller product.
In
sixth grade, students studied the relationship between ratio/rate and
multiplication/division. Students’ use and understanding
of fractions, decimals, and percents in seventh grade should include the
application of ratio, rates, and proportions to discounts, taxes, tips,
interest, unit costs, and similar shapes in problem solving situations. Problem
solving opportunities should include the use of graphs, tables, and equations.
Instruction should focus on the conceptual understanding of the operation
involved rather than the procedure involved. Students may need to review the
differences in ratios and rates. Students should determine the reasonableness
of solutions and demonstrate an understanding of the magnitude of the numbers
involved and the computational meaning of the operation being preformed when
justifying their results. Problems found in daily newspapers, magazines, and on
television make the problems applicable to the student’s own world.
Connections to:
Other Seventh Grade Indicators
Other Indicators
7-4.7 Explain the proportional relationship among
attributes of similar shapes.
7-4.8 Apply proportional reasoning to find missing
attributes of similar shapes.
7-5.1 Use ratio and
proportion to solve problems involving scale factors and rates.
The above indicators provide connections for i-depth exploration
of the concept of proportional reasoning from a geometry perspective. It is
important for seventh grade students to extend their understanding of ratio and
proportion as they explore similarity. Seventh grade students combine this
knowledge with knowledge of proportionality to find measures of missing sides,
beginning with explorations of various types of similar shapes. Students must
understand that pairs of similar shapes have proportional perimeters and side
lengths. Problems that involve creating and analyzing scale drawings give
students good experience with similarity and proportionality.
Teacher Notes:
Remediation
Strategies
“Part-Part-Whole”
(PPW) is an effective strategy for solving addition and subtraction problems.
This may be a helpful strategy for students as they work to generate strategies
to add and subtract integers. Students may have been exposed to this approach
in previous grades. Therefore using refocusing questions that help students
identify the “Parts” versus the “Whole” may be sufficient to move the student
forward. For students who have not used or are not comfortable with a PPW
strategy, more assistance may be needed.
There are
many variations to the PPW strategy – “Join”, “Separate”, “Compare” to name a
few. However, for a student that is experiencing difficulty, using multiple
variations may not be the best approach. In the PPW strategy, the parts equal
the whole. All addition and subtraction type problems are approached by asking
what the parts are and what the whole is. Identification of the parts and the
whole is where a student may experience the most difficulty. If that is the
case, substituting his/her final answer in the PPW “formula” will enable the
student to quickly see their error. In the following examples, addition or
subtraction may be used to solve the problem depending on what is unknown. But
in all examples, a simple PPW strategy was used.