Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators High School - for Students with the Most Significant ...
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NEBRASKA
Alternate Mathematics
Instructional Supports
for
NSCAS Mathematics
Extended Indicators
High School
for
Students with the Most Significant Cognitive Disabilities
who take the
Statewide Mathematics Alternate AssessmentTable of Contents
Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
The Role of Extended Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Students with the Most Significant Intellectual Disabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Alternate Assessment Determination Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Instructional Supports Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Mathematics—Grade 11
MA 11.1 Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
MA 11.1.1 Numeric Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
MA 11.1.1.a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
MA 11.1.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
MA 11.1.2.a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
MA 11.1.2.b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
MA 11.1.2.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Mathematics—Grade 11
MA 11.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
MA 11.2.1 Algebraic Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
MA 11.2.1.b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
MA 11.2.1.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
MA 11.2.1.e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
MA 11.2.1.g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
MA 11.2.2 Algebraic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
MA 11.2.2.a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
MA 11.2.2.d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
MA 11.2.2.e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
MA 11.2.2.g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
MA 11.2.2.h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
ii Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMathematics—Grade 11
MA 11.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
MA 11.3.1 Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
MA 11.3.1.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
MA 11.3.1.d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
MA 11.3.2 Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
MA 11.3.2.b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
MA 11.3.2.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
MA 11.3.2.d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
MA 11.3.2.e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
MA 11.3.2.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
MA 11.3.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
MA 11.3.3.d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
MA 11.3.3.e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Mathematics—Grade 11
MA 11.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
MA 11.4.2 Analysis and Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
MA 11.4.2.a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
MA 11.4.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
MA 11.4.3.b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
MA 11.4.3.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School iiiOverview Introduction Mathematics standards apply to all students, regardless of age, gender, cultural or ethnic background, disabilities, aspirations, or interest and motivation in mathematics (NRC, 1996). The mathematics standards, extended indicators, and instructional supports in this document were developed by Nebraska educators to facilitate and support mathematics instruction for students with the most significant intellectual disabilities. They are directly aligned to the Nebraska’s College and Career Ready Standards for Mathematics adopted by the Nebraska State Board of Education. The instructional supports included here are sample tasks that are available to be used by educators in classrooms to help instruct students with significant intellectual disabilities. The Role of Extended Indicators For students with the most significant intellectual disabilities, achieving grade‑level standards is not the same as meeting grade‑level expectations, because the instructional program for these students addresses extended indicators. It is important for teachers of students with the most significant intellectual disabilities to recognize that extended indicators are not meant to be viewed as sufficient skills or understandings. Extended indicators must be viewed only as access or entry points to the grade‑level standards. The extended indicators in this document are not intended as the end goal but as a starting place for moving students forward to conventional reading and writing. Lists following “e.g.” in the extended indicators are provided only as possible examples. Students with the Most Significant Intellectual Disabilities In the United States, approximately 1% of school‑aged children have an intellectual disability that is “characterized by significant impairments both in intellectual and adaptive functioning as expressed in conceptual, social, and practical adaptive domains” (U.S. Department of Education, 2002 and American Association of Intellectual and Developmental Disabilities, 2013). These students show evidence of cognitive functioning in the range of severe to profound and need extensive or pervasive support. Students need intensive instruction and/or supports to acquire, maintain, and generalize academic and life skills in order to actively participate in school, work, home, or community. In addition to significant intellectual disabilities, students may have accompanying communication, motor, sensory, or other impairments. Alternate Assessment Determination Guidelines The student taking a Statewide Alternate Assessment is characterized by significant impairments both in intellectual and adaptive functioning which is expressed in conceptual, social, and practical adaptive domains and that originates before age 18 (American Association of Intellectual and Developmental Disabilities, 2013). It is important to recognize the huge disparity of skills possessed by students taking an alternate assessment and to consider the uniqueness of each child. 4 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School
Thus, the IEP team must consider all of the following guidelines when determining the
appropriateness of a curriculum based on Extended Indicators and the use of the Statewide Alternate
Assessment.
● The student requires extensive, pervasive, and frequent supports in order to acquire,
maintain, and demonstrate performance of knowledge and skills.
● The student’s cognitive functioning is significantly below age expectations and has an impact
on the student’s ability to function in multiple environments (school, home, and community).
● The student’s demonstrated cognitive ability and adaptive functioning prevent completion
of the general academic curriculum, even with appropriately designed and implemented
modifications and accommodations.
● The student’s curriculum and instruction is aligned to the Nebraska College and Career
Ready Mathematics Standards with Extended Indicators.
● The student may have accompanying communication, motor, sensory, or other impairments.
The Nebraska Department of Education’s technical assistance documents “IEP Team Decision
Making Guidelines—Statewide Assessment for Students with Disabilities” and “Alternate
Assessment Criteria/Checklist” provide additional information on selecting appropriate statewide
assessments for students with disabilities. School Age Statewide Assessment Tests for Students with
Disabilities—Nebraska Department of Education.
Instructional Supports Overview
The mathematics instructional supports are scaffolded activities available for use by educators who
are instructing students with significant intellectual disabilities. The instructional supports are aligned
to the extended indicators in grades three through eight and in high school. Each instructional support
includes the following components:
● Scaffolded activities for the extended indicator
● Prerequisite extended indicators
● Key terms
● Additional resources or links
The scaffolded activities provide guidance and suggestions designed to support instruction with
curricular materials that are already in use. They are not complete lesson plans. The examples and
activities presented are ready to be used with students. However, teachers will need to supplement
these activities with additional approved curricular materials. The scaffolded activities adhere
to research that supports instructional strategies for mathematics intervention, including explicit
instruction, guided practice, student explanations or demonstrations, visual and concrete models, and
repeated, meaningful practice.
Each scaffolded activity begins with a learning goal, followed by instructional suggestions that are
indicated with the inner level, circle bullets. The learning goals progress from less complex to more
complex. The first learning goal is aligned with the extended indicator but is at a lower achievement
level than the extended indicator. The subsequent learning goals progress in complexity to the last
learning goal, which is at the achievement level of the extended indicator.
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 5The inner level, bulleted statements provide instructional suggestions in a gradual release model. The first one or two bullets provide suggestions for explicit, direct instruction from the teacher. From the teacher’s perspective, these first suggestions are examples of “I do.” The subsequent bullets are suggestions for how to engage students in guided practice, explanations, or demonstrations with visual or concrete models, and repeated, meaningful practice. These suggestions start with “Ask students to . . .” and are examples of moving from “I do” activities to “we do” and “you do” activities. Visual and concrete models are incorporated whenever possible throughout all activities to demonstrate concepts and provide models that students can use to support their own explanations or demonstrations. The prerequisite extended indicators are provided to highlight conceptual threads throughout the extended indicators and show how prior learning is connected to new learning. In many cases, prerequisites span multiple grade levels and are a useful resource if further scaffolding is needed. Key terms may be selected and used by educators to guide vocabulary instruction based on what is appropriate for each individual student. The list of key terms is a suggestion and is not intended to be an all‑inclusive list. Additional links from web‑based resources are provided to further support student learning. The resources were selected from organizations that are research based and do not require fees or registrations. The resources are aligned to the extended indicators, but they are written at achievement levels designed for general education students. The activities presented will need to be adapted for use with students with significant intellectual disabilities. 6 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School
Mathematics—Grade 11
MA 11.1 Number
MA 11.1.1 Numeric Relationships
MA 11.1.1.a
Compare and contrast subsets of the complex number system, including imaginary, rational,
irrational, integers, whole, and natural numbers.
Extended: Sort fractions, decimals, and whole numbers by type (e.g., _5 , 4, 1.7).
3
Scaffolding Activities for the Extended Indicator
Identify differences between whole-number, fraction, and decimal forms.
● Use a chart to show the differences in the written form for whole numbers, fractions, and
decimals. For example, create a three‑column chart with the columns labeled Whole
Number, Fraction, and Decimal with examples of each number form.
Whole Number Fraction Decimal
_
1 _
2 __
7
5 68 135 2 5 10 0.5 6.9 12.7
List defining characteristics for each number form.
Whole Number Fraction Decimal
_
1 _
2 __
7
5 68 135 2 5 10 0.5 6.9 12.7
part of a whole
counting numbers and 0 part of a whole
top number
no decimal point number/decimal/number
bottom number
● Ask students to identify whether a given value meets the criteria (defining characteristics) of
a whole number, fraction, or decimal.
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 7MA 11.1.1 Numeric Relationships
Sort whole numbers, fractions, and decimals by number form.
● Use a chart and a collection of cards with whole numbers, fractions, and decimals to show
how to sort the cards by the number form. For example, use a three‑column chart as
shown. Draw a card from the collection of cards and ask clarifying questions about the card.
Demonstrate sorting the card into the correct category based on the answers to the clarifying
questions. For example, does the number have a decimal point?
Whole Number Fraction Decimal
_
3
35 4 29.7
● Ask students to sort cards with whole numbers, fractions, and decimals into categories by
number form.
Prerequisite Extended Indicator
MAE 8.1.1.a—Distinguish among whole numbers, fractions, and decimals.
Key Terms
decimal, decimal point, fraction, whole number
Additional Resources or Links
https://www.mathlearningcenter.org/sites/default/files/documents/sample_materials/br3‑tg‑u4‑m3.
pdf
https://www.mathlearningcenter.org/sites/default/files/documents/sample_materials/br4‑tg‑u3‑m3.
pdf
8 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.1.2 Operations
MA 11.1.2 Operations
MA 11.1.2.a
Compute with subsets of the complex number system, including imaginary, rational, irrational,
integers, whole, and natural numbers.
Extended: Add and subtract two-digit numbers with regrouping.
Scaffolding Activities for the Extended Indicator
Add a two-digit number to a one-digit number with regrouping.
● Use a place value mat and base ten blocks to teach regrouping values of 10 or more into
tens and ones. Compare 10 unit cubes to a tens rod to demonstrate that 10 unit cubes are
equal to the tens rod.
=
Next, demonstrate that 18 unit cubes are equal to 1 tens rod and 8 unit cubes. First, place
18 unit cubes in the ones column. Next, exchange 10 unit cubes for a tens rod. Continue
regrouping values of 10 or more to demonstrate the exchange of 10 unit cubes for a
tens rod.
Tens Ones Tens Ones Tens Ones
● Ask students to regroup values of 10 or more into tens and ones using a place value mat
and base ten blocks.
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 9MA 11.1.2 Operations
● Use a place value mat and base ten blocks to teach addition with regrouping. Present the
problem 46 + 9 = . Demonstrate the value 46 on the place value mat. Then place
9 unit cubes in the ones column in the lower half to represent the value of the second
addend.
Tens Ones
Indicate that the total in the ones column is larger than 10. Demonstrate that 10 unit cubes
can be exchanged for a tens rod, and that rod will be placed in the tens column.
Tens Ones
Indicate that there are now 5 tens and 5 ones, which is written as 55. Complete the number
sentence 46 + 9 = 55.
10 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.1.2 Operations
● Ask students to add a one‑digit number to a two‑digit number with regrouping using a place
value mat and base ten blocks. Use sticky notes labeled with tens and ones (left and right)
to teach addition with regrouping using the standard algorithm. Present the problem 37 + 6
in standard vertical form on a piece of oversize grid paper with the tens column and the ones
column labeled.
Tens Ones
3 7
+ 6
Add the digits in the ones column. Write the number 13 on a sticky note that has been
labeled with a tens column and a ones column.
Tens Ones
1 3
Cut the sticky note in half. Place the right half under the ones column of the problem. Place
the left half above the tens column of the problem.
Tens
1
Tens Ones
3 7
+ 6
Ones
3
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 11MA 11.1.2 Operations
Add the tens column to complete the problem, showing that 37 + 6 = 43. Gradually fade out
the use of cutting the sticky note and placing the two halves to using the information on the
sticky note to record the ones value in the answer and the tens value above the tens column.
Transition next to leaving the column labels off the sticky note. Finally, omit the column
labels on the problem and use the sticky note as scratch paper.
● Ask students to use a regrouping strategy to add a one‑digit number to a two‑digit number.
Subtract a one-digit number from a two-digit number with regrouping.
● Demonstrate that 15 (1 ten and 5 ones) can also be represented as 15 ones. First, place
1 tens rod and 5 unit cubes on a place value mat. Next, exchange the tens rod for 10 unit
cubes and place all the unit cubes under the ones column. Continue to demonstrate with
numbers such as 24 and 37.
Tens Ones Tens Ones Tens Ones
● Use a place value mat and base ten blocks to teach subtraction with regrouping. Present the
problem 25 – 9 = . Represent 25 on a place value mat with base ten blocks. Present
the dilemma of needing to take away 9 ones and having only 5. Demonstrate exchanging
a tens rod for 10 unit cubes. Demonstrate removing (subtracting) 9 from the 15 unit cubes.
Count the remaining rods and blocks to find the solution to the problem. Complete the
number sentence: 25 – 9 = 16.
Tens Ones Tens Ones Tens Ones Tens Ones
● Ask students to subtract a one‑digit number from a two‑digit number with regrouping using a
place value mat and base ten blocks.
12 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.1.2 Operations
● Use oversize grid paper with a tens column and a ones column labeled to teach subtraction
of a one‑digit number from a two‑digit number with regrouping. For example, present the
problem 72 – 5 = . Next, present the dilemma of having only 2 ones and needing
to take away 5. Demonstrate regrouping. Finally, subtract the ones column and the tens
column to find the solution of the problem. Complete the number sentence: 72 – 5 = 67.
Tens Ones Tens Ones Tens Ones
7 2 6 10 + 2 = 12 6 12
− 5 − 5 − 5
6 7
● Ask students to use a grid and a regrouping strategy to subtract a one‑digit number from a
two‑digit number.
Prerequisite Extended Indicators
MAE 4.1.1.a—Identify representations of numbers 0–100.
MAE 3.1.2.a—Add and subtract, through 20 without regrouping.
Key Terms
addition, exchange, regrouping, subtraction
Additional Resources or Links
https://www.engageny.org/resource/grade‑4‑mathematics‑module‑1‑topic‑d‑lesson‑11
https://www.engageny.org/resource/grade‑4‑mathematics‑module‑1‑topic‑e‑lesson‑13
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 13MA 11.1.2 Operations
MA 11.1.2.b
Simplify expressions with rational exponents.
Extended: Rewrite a repeated multiplication problem as an exponential expression with a
whole number base and a whole number exponent (e.g., 3 × 3 × 3 × 3 = 34).
Scaffolding Activities for the Extended Indicator
Recognize the vocabulary used to name exponents.
● Use an expression to explain how to read and write exponents. For example, in the
expression shown, the number 5 represents the base and the number 4 represents the
exponent. It is read as “five to the fourth power.”
Continue to read examples of exponential expressions, like 63, and 42. Indicate that the
base is read first, then the exponent: “six to the third power” or “six cubed” and “four to the
second power” or “four squared.”
● Ask students to select the correct exponential expression when given a set of three
expressions with the same exponents but different bases. For example, present the choices
42, 22, and 32 and ask which expression shows two squared or which expression shows
three to the second power.
● Ask students to select the correct exponential expression when given a set of three
expressions with different exponents but the same bases. For example, present the choices
43, 42, and 44 and ask which expression shows four cubed or which expression shows four to
the second power.
Rewrite a repeated multiplication problem as an exponential expression.
● Demonstrate that when a number is multiplied by itself several times, it may be written as
an exponential expression. For example, 5 can be multiplied by itself 4 times. Represent the
base and the exponent as shown.
Continue to demonstrate with other examples by writing and describing correct usage of the
base and exponent.
● Ask students to select the exponential expression represented by a given multiplication
problem when presented options with the same base and different exponents. For example,
ask students, “Which expression is equivalent to 7 × 7 × 7?” Present 72, 73, and 74 as the
answer options. Students should select 73 as the correct answer.
14 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.1.2 Operations
● Ask students to select the exponential expression represented by a given multiplication
problem when presented options with different bases and different exponents. For example,
ask students, “Which expression is equivalent to 7 × 7 × 7?” Present 57, 73, and 37 as the
answer options. Students should select 73 as the correct answer.
Prerequisite Extended Indicators
MAE 8.1.1.b—Represent numbers with the base of 2, 3, 4, or 5 and positive exponents of 2 and
3 in expanded form (e.g., 43 = 4 × 4 × 4).
MAE 6.1.1.b—Represent 10, 100, 1,000, or 10,000 as a power of 10.
Key Terms
base, cubed, exponent, power, squared
Additional Resources or Links
https://curriculum.illustrativemathematics.org/MS/teachers/3/7/1/preparation.html
https://www.engageny.org/resource/grade‑8‑mathematics‑module‑1‑topic‑lesson‑1/file/46091
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 15MA 11.1.2 Operations
MA 11.1.2.c
Select, apply, and explain the method of computation when problem solving using real numbers
(e.g., models, mental computation, paper‑pencil, or technology).
Extended: Given a real-world problem, identify an operation that leads to a solution.
Scaffolding Activities for the Extended Indicator
Identify an operation that leads to the solution of a real-world problem.
● Use story problems to demonstrate how to look for key words or other clues about the
operation needed to solve the problem. Present the story problem shown below. Explain that
words like “altogether,” “total,” “more,” and “sum” all indicate addition. The question in this
problem needs addition to solve the problem.
The soccer team scored 1 point in the first half of the game.
In the second half of the game, the soccer team scored 5 points.
How many points did the soccer team score altogether?
Present the story problem shown below. Explain that words like “remain,” “left over,” “less,”
and “difference” all indicate subtraction. This problem about moths needs subtraction to
solve.
There are 10 moths flying around a light bulb at night.
Then 3 moths fly away.
How many moths remain?
Present the story problem shown below. Explain that words like “each,” “in a row,” and
“product” indicate multiplication. This problem about reading needs multiplication to solve.
Micah reads 20 pages of a book each day.
Micah reads for 5 days in a row.
How many pages of the book does Micah read?
Present the story problem shown below. Explain that words like “each time,” “per,” and
“quotient” indicate division. This problem about swimming needs division to solve.
Rachel is swimming laps and wants to swim 40 laps this week.
She swims 4 times this week and swims an equal number of laps each time.
How many laps should Rachel swim each time to complete 40 laps?
Continue to demonstrate identifying the correct operation (+, –, ×, or ÷) in a variety of story
problems. It might be helpful to create a chart with the four operations and clue words as
a reference. Provide drawings as necessary to reinforce the operation needed to solve the
problem.
16 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.1.2 Operations
● Ask students to identify the operation needed to solve a story problem when given a drawing
or chart with clue words to reference.
● Ask students to identify the operation that leads to the solution of a real‑world problem.
Identify an expression or equation that leads to the solution of a real-world problem.
● Use story problems to demonstrate how to identify an expression or equation that leads to
the solution. For example, when a real‑world problem indicates that there will be more of
something (or the answer will be a greater number than what is in the problem), it typically
means it will either be an addition or multiplication expression.
Present the following story problem about a basketball game.
Stef made 4 shots in her basketball game.
Each shot Stef made was worth 3 points.
How many points did Stef score?
Demonstrate identifying the expression or equation by first finding the numbers in the story
problem that will be needed for the expression. For this example, explain that the numbers
are 4 and 3, representing the shots and the point value. Next, look for the clue words that
indicate which operation to use with the 4 and 3. In this case, multiplication will be used,
so the expression can be 4 × 3 or 3 × 4. Addition can be ruled out as an option because
4 + 3 in this context would be adding the number of shots to the points, so the answer
would not match the question, which only asks about points. Another expression could be
3 + 3 + 3 + 3, but 4 × 3 is the simpler option.
Present the following story problem about apples.
There are 30 slices of apples for a snack.
The slices of apples will be shared equally with 5 friends.
How many apple slices will each friend get?
For this example, first identify that the numbers in the problem are 30 and 5. Next, explain
that the answer to the problem will be less than the original amount of 30 slices, so the
problem either involves subtraction or division. Reference the clue words “shared equally”
in this context. It may help to demonstrate with manipulatives. The expression to best fit this
example is 30 ÷ 5.
Continue to demonstrate identifying the expression and the equation that leads to a solution
in a variety of one‑step addition, subtraction, multiplication, and division story problems.
● Ask students to identify an expression that leads to the solution of a real‑world problem
when given a choice of two or more expressions.
● Ask students to identify an equation that leads to the solution of a real‑world problem when
given a choice of two or more equations.
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 17MA 11.1.2 Operations
Prerequisite Extended Indicators
MAE 6.2.1.a—Match a simple word phrase with an input/output box.
MAE 5.1.2.b—Divide a two‑digit whole number by a single‑digit number with no remainder.
MAE 3.2.3.a—Solve a one‑step real‑world problem using addition or subtraction 0–9.
MAE 3.2.1.b—Identify a multiplication equation as representing equal groups up to 20.
Key Terms
add, divide, multiply, operation, solution, subtract
Additional Resources or Links
https://nysed‑prod.engageny.org/resource/grade‑2‑mathematics‑module‑4‑topic‑c‑lesson‑16
https://nysed‑prod.engageny.org/resource/grade‑6‑mathematics‑module‑4‑topic‑f‑lesson‑18
18 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMathematics—Grade 11
MA 11.2 Algebra
MA 11.2.1 Algebraic Relationships
MA 11.2.1.b
Analyze a relation to determine if it is a function given graphs, tables, or algebraic notation.
Extended: Identify a graph that represents a given linear function from a table.
Scaffolding Activities for the Extended Indicator
Identify ordered pairs from a table.
● Use a table to show how to form ordered pairs. For example, the input values represent the
x‑coordinates and the output values represent the y‑coordinates in the ordered pair (x, y).
So, the ordered pairs for the relationship from the table are (1, 7), (2, 8), and (3, 9).
Input Output
1 7
2 8
3 9
● Ask students to identify ordered pairs from a given table. Students should identify the
ordered pairs as (2, 4), (4, 3), and (6, 2).
x y
2 4
4 3
6 2
Identify a graph that represents a given linear function from a table.
● Explain the connection between ordered pairs from a table and a graph on a coordinate
plane. For example, the table shown has the ordered pairs (2, 1), (4, 2), and (6, 3).
x y
2 1
4 2
6 3
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 19MA 11.2.1 Algebraic Relationships
Since the ordered pairs indicate the x‑ and y‑coordinates of three points on a graph, show
the connection by graphing those three points.
y
6
5
4
3
2
1
x
0 1 2 3 4 5 6
Show other tables with at least three points on a line, and then show how to identify the
graph that corresponds to each table.
● Ask students to identify a graph that represents a given linear function from a table. For
example, present the following table.
Day Miles
1 1
2 1
3 1
Give students three possible graphs to choose from.
y y y
5 5 5
4 4 4
Miles
Miles
Miles
3 3 3
2 2 2
1 1 1
x x x
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
Day Day Day
Students should identify that the points in the middle graph represent the same ordered
pairs that are in the table.
20 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.2.1 Algebraic Relationships
Prerequisite Extended Indicators
MAE 11.2.1.c—Identify a linear function from a graph.
MAE 8.2.1.d—Given a graph of a line through the origin and a point on the line, determine
another point on the line.
MAE 6.3.2.c—Identify a point on a 4‑by‑4 grid in quadrant 1.
Key Terms
coordinate plane, function, graph, linear, ordered pair, relationship, table
Additional Resources or Links
https://nysed‑prod.engageny.org/resource/grade‑8‑mathematics‑module‑6‑topic‑lesson‑1/
file/48541
https://curriculum.illustrativemathematics.org/MS/teachers/1/6/17/index.html
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 21MA 11.2.1 Algebraic Relationships
MA 11.2.1.c
Classify a function given graphs, tables, or algebraic notation, as linear, quadratic, or neither.
Extended: Identify a linear function from a graph.
Scaffolding Activities for the Extended Indicator
Identify a linear function from a graph.
● Use sketches of graphs to show linear functions. Make note that each graph is a line that
is perfectly straight. This can be verified by using a straight object, like a pencil or ruler, to
follow the line.
y y
x x
y y
x x
22 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.2.1 Algebraic Relationships
Also show graphs of functions that are NOT linear. Indicate that when a graph of a function
has curves or corners it is not a linear function. Linear functions can also be called linear
relationships since they show the relationship between the x‑ and y‑values on a graph.
y y
x x
y y
x x
● Ask students to identify a linear function from a graph. For example, give students the
following graphs.
y y y
x x x
Ask students to choose the graph that shows the linear relationship. Students should choose
the graph on the left.
Prerequisite Extended Indicators
MAE 5.3.2.b—Identify the x‑ or y‑coordinate of whole‑numbered points in quadrant I.
Key Terms
curve, function, graph, line, linear, straight
Additional Resources or Links
https://www.map.mathshell.org/download.php?fileid=1740
https://nysed‑prod.engageny.org/resource/grade‑8‑mathematics‑module‑5‑topic‑lesson‑5/
file/48311
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 23MA 11.2.1 Algebraic Relationships
MA 11.2.1.e
Analyze and graph linear functions and inequalities (point‑slope form, slope‑intercept form, standard
form, intercepts, rate of change, parallel and perpendicular lines, vertical and horizontal lines, and
inequalities).
Extended: Given an x-, y-table of values, determine if the graph of the values forms a
horizontal line or a vertical line.
Scaffolding Activities for the Extended Indicator
Identify a horizontal line and a vertical line.
● Use real‑world objects and images to teach the meaning of horizontal and vertical. For
example, the top of a table is horizontal, and the legs of a table are vertical.
● Ask students to identify horizontal and vertical lines using real‑world objects and images.
Determine whether a graph of values from a table forms a horizontal line or a vertical line.
● Use a coordinate plane to show horizontal and vertical lines. For example, the x‑axis is a
horizontal line and the y‑axis is a vertical line.
y
vertical
6
5
4
3
2
1 horizontal
x
0 1 2 3 4 5 6
In addition to the axes being examples of a horizontal line and a vertical line, a graph of a
line can also be a horizontal line or a vertical line. An example of a horizontal line graph is
shown.
y
6
5
4
3
2
1
x
0 1 2 3 4 5 6
Show a variety of graphs of lines and identify which are horizontal lines and which are
vertical lines.
24 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.2.1 Algebraic Relationships
● Use a table to demonstrate that ordered pairs can be formed to represent the relationship in
the table and then graphed.
x y
3 3
3 4
3 5
The ordered pairs for this table are (3, 3), (3, 4), and (3, 5) and may be graphed as shown.
y
6
5
4
3
2
1
x
0 1 2 3 4 5 6
Note that since all the x‑coordinates from the table are 3, a straight line can be drawn
through the three points given, so the relationship in the table is represented by a vertical
line graph. Show a variety of tables, some with horizontal line graphs and some with vertical
line graphs. Make sure to note that a table with all the same values for each x‑coordinate will
make a vertical line and a table with all the same values for each y‑coordinate will make a
horizontal line.
● Ask students to determine if a given table will have a graph that is a horizontal line or a
vertical line.
Prerequisite Extended Indicators
MAE 6.3.2.c—Identify a point on a 4‑by‑4 grid in quadrant 1.
MAE 11.2.1.c—Identify a linear function from a graph.
MAE 11.2.1.b—Identify a graph that represents a given linear function from a table.
Key Terms
coordinate plane, graph, horizontal, ordered pair, table, vertical, x‑coordinate, y‑coordinate
Additional Resources or Links
https://www.engageny.org/resource/grade‑5‑mathematics‑module‑6‑topic‑lesson‑5/file/69636
http://tasks.illustrativemathematics.org/content‑standards/5/G/A/1/tasks/489
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 25MA 11.2.1 Algebraic Relationships
MA 11.2.1.g
Analyze and graph quadratic functions (standard form, vertex form, finding zeros, symmetry,
transformations, determine intercepts, and minimums or maximums).
Extended: Use the graph of a linear function to locate the ordered pair where y = 0.
Scaffolding Activities for the Extended Indicator
Identify the location where a graph of a linear function intersects the x-axis.
● Use a coordinate graph to demonstrate where the x‑axis, y‑axis, and origin are located.
Explain that the ordered pair (0, 0) indicates that the origin is where the x‑ and y‑coordinates
are both 0 because ordered pairs are always written as (x, y).
26 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.2.1 Algebraic Relationships
● Use the graph of a linear function to demonstrate how to locate where the function intersects
the x‑axis. Start with locating the origin, and then follow the x‑axis either left or right to where
it crosses the linear function. Indicate that intersection by circling the point or highlighting it
in some way. Follow this same process with a variety of graphs of linear functions, each time
finding where the line intersects the x‑axis. Use graphs that cross to the left of the origin as
well as to the right.
● Ask students to identify the location where a graph of a linear function intersects the x‑axis.
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 27MA 11.2.1 Algebraic Relationships
Use the graph of a linear function to locate the ordered pair where y = 0.
● Use the graph of a linear function to demonstrate how to locate the ordered pair where y = 0,
which is where the graph intersects the x‑axis.
Since the y‑coordinate at the point where the graph crosses the x‑axis is 0, the ordered pair
of that location can be written as (x, 0), where x represents the x‑coordinate. In the graph
shown, the ordered pair is (2, 0). Follow this same process with a variety of linear functions,
including those that are graphed with only points, and no line, as shown.
Explain that the location where the function intersects the x‑axis is still shown by the ordered
pair (x, 0) whether it is a line or points in a line. For this graph, it is at the point (3, 0).
● Ask students to use the graph of a linear function to locate the ordered pair where y = 0.
28 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.2.1 Algebraic Relationships
Prerequisite Extended Indicators
MAE 11.2.1.c—Identify a linear function from a graph.
MAE 8.2.1.d—Given a graph of a line through the origin and a point on a line, determine another
point on the line.
MAE 6.3.2.c—Identify a point on a 4‑by‑4 grid in quadrant 1.
Key Terms
function, linear, ordered pair, origin, x‑axis, x‑coordinate, y‑axis, y‑coordinate
Additional Resources or Links
https://curriculum.illustrativemathematics.org/k5/teachers/grade‑5/unit‑7/lesson‑2/lesson.html
https://www.engageny.org/resource/grade‑6‑mathematics‑module‑3‑topic‑c‑lesson‑14
https://www.engageny.org/resource/grade‑7‑mathematics‑module‑1‑topic‑lesson‑5
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 29MA 11.2.2 Algebraic Processes
MA 11.2.2 Algebraic Processes
MA 11.2.2.a
Convert equivalent rates (e.g., miles per hour to feet per second).
Extended: Convert equivalent rate using money.
Scaffolding Activities for the Extended Indicator
Convert equivalent rates for single coins or one bill up to $1.00.
● Describe coins as each having a particular value. For example, a penny is worth 1¢, a nickel
is worth 5¢, a dime is worth 10¢, and a quarter is worth 25¢. Use coins or coin manipulatives
to show examples.
1¢ 5¢ 10¢ 25¢
Penny Nickel Dime Quarter
● Explain that different coins may be used to create the same value as another coin. Present
the table shown and demonstrate making conversions between coins and 1 dollar.
penny nickel dime quarter dollar
penny 1 penny
nickel 5 pennies 1 nickel
dime 10 pennies 2 nickels 1 dime
quarter 25 pennies 5 nickels 1 quarter
dollar 100 pennies 20 nickels 10 dimes 4 quarters 1 dollar
Use various examples to show the equivalencies and use skip‑counting as a strategy.
= =
● Ask students to convert a single larger monetary value into coins with smaller values. For
example, ask students to answer the following questions.
How many dimes are there in 1 dollar? How many nickels are there in 1 quarter?
How many pennies are there in 1 dime? How many quarters are there in 1 dollar?
● Ask students to convert coins with smaller values into a single coin with a larger value. For
example, ask students to answer the following questions.
What single coin is equal to 5 nickels? What single coin is equal to 10 pennies?
What single coin is equal to 25 pennies? What single coin is equal to 2 nickels?
30 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.2.2 Algebraic Processes
Convert equivalent rates for multiple coins or bills up to $2.00.
● Explain that the value of multiple coins or bills may be found using the value of single coins
or bills. For example, quarters can be used to make 2 dollars because if it is known that
4 quarters are equal to 1 dollar, then 8 quarters must be equal to 2 dollars.
=
=
Demonstrate the same process to find the value of multiple coins. For example, if there are
5 nickels in 1 quarter, then there are 10 nickels in 2 quarters.
= =
● Ask students to convert multiple coins or bills. For example, ask students to answer the
following questions.
How many nickels are there in 2 dollars? How many pennies are there in 2 quarters?
How many nickels are there in 2 dimes? How many dimes are there in 2 dollars?
● Ask students to convert coins or bills in multiple ways. For example, ask students, “What are
some coins that are equal to 2 dollars?” Students may respond with 20 dimes, 8 quarters,
40 nickels, etc.
Prerequisite Extended Indicators
MAE 8.2.1.b—Describe the rate of change of a proportional relationship given a table.
MAE 7.2.1.b—Identify a ratio between two quantities using a model.
MAE 6.2.2.f—Find a missing number in a table with the ratio of 1:2, 1:3, or 1:10.
Key Terms
convert, dime, dollar bill, exchange, nickel, penny, quarter
Additional Resources or Links
http://nlvm.usu.edu/en/nav/frames_asid_325_g_3_t_1.html?from=search.html?qt=money
(Note: Java required for website. Most recent version recommended, but not needed.)
https://www.engageny.org/resource/grade‑2‑mathematics‑module‑7‑topic‑b‑lesson‑6
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 31MA 11.2.2 Algebraic Processes
MA 11.2.2.d
Perform operations on rational expressions (add, subtract, multiply, divide, and simplify).
Extended: Add two linear expressions (e.g., (2x + 1) + (3x + 2) = 5x + 3.
Scaffolding Activities for the Extended Indicator
Identify and sort like terms.
● Present an addition expression using groups of similar objects.
+ + +
4 balls + 2 skateboards + 2 balls + 2 skateboards
Then place the balls next to each other and the skateboards next to each other to show
(without solving yet) how the similar objects can be grouped together or placed next to each
other in the addition expression.
+ + +
4 balls + 2 balls + 2 skateboards + 2 skateboards
● Use objects from the classroom (pens, markers, crayons, paper clips, etc.) to
create addition expressions and model rewriting the addition expressions by placing
the similar items next to each other in the expression. For example, start with
3 paper clips + 5 pens + 4 paper clips + 2 pens. Then place the similar objects next to each
other to create the number expression 3 paper clips + 4 paper clips + 5 pens + 2 pens.
32 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.2.2 Algebraic Processes
● Explain that addition expressions can be written with numbers and variables. A variable is
a letter that stands for an unknown number. The addition expression 5x + 4 + 3x + 2 has
numbers and variables. Since the terms 5x and 3x both include an x, they are called like
terms and can be placed next to each other in a math expression. The whole numbers
4 and 2 can be placed next to each other because neither has a variable and they are also
like terms. The expression can be rewritten as 5x + 3x + 4 + 2.
● Model sorting expressions into adjacent like terms.
4w + 3 + 5w + 2 6y + 4 + 2y + 5 7x + 5 + 2x + 3 5m + 8 + 3m + 1
● Ask students to sort expressions into adjacent like terms.
Add two linear expressions.
● Demonstrate adding or combining similar objects to find the total number of objects. For
example, expand on the previous demonstration using balls and skateboards to find the total
number of balls and the total number of skateboards.
+ + +
4 balls + 2 balls + 2 skateboards + 2 skateboards
= 6 balls and 4 skateboards
Explain that after the similar objects are placed next to each other in the addition expression,
the similar objects can be added together.
● Use objects from the classroom to model creating addition expressions, rewriting addition
expressions by placing like objects next to each other, and then combining to find the
total number of like objects. For example, start with 4 markers + 3 erasers + 5 markers
+ 2 erasers. Then place the similar objects next to each other and rewrite the addition
expression as 4 markers + 5 markers + 3 erasers + 2 erasers. Lastly, combine (or add) the
similar objects to show 9 markers + 5 erasers.
● Explain that, just like similar objects can be added or combined to find the total number of
objects, addition expressions with variables can be combined when they have like terms.
For example, 8c and 2c are like terms because they both use the variable c. The addition
expression 6w + 3 + 5w + 2 is an example of an addition expression with like terms. The
terms can be sorted and then added together. The addition expression can be rewritten as
6w + 5w + 3 + 2 so the like terms are next to each other. Then the like terms can be added
together. Demonstrate that 6w + 5w = 11w and 3 + 2 = 5, so the addition expression can be
rewritten as 11w + 5.
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 33MA 11.2.2 Algebraic Processes
● Model adding linear expressions.
3w + 5 + 4w + 2 2y + 9 + 7y + 4 8x + 1 + 3x + 5 7m + 6 + 2m + 2
● Ask students to add linear expressions.
Prerequisite Extended Indicators
MAE 7.2.2.b—Identify equivalent expressions with one variable (2n + 3n is the same as 5n).
MA 3.1.2.a—Add and subtract, through 20 without regrouping.
Key Terms
addition, like term, number sentence, similar, variable
Additional Resources or Links
https://www.engageny.org/resource/grade‑7‑mathematics‑module‑3‑topic‑lesson‑1
http://tasks.illustrativemathematics.org/content‑standards/6/EE/A/4/tasks/461
34 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.2.2 Algebraic Processes
MA 11.2.2.e
Evaluate expressions at specified values of their variables (polynomial, rational, radical, and absolute
value).
Extended: Evaluate a linear expression at a specified value of the variable. Include cases
where combining like terms or using the distributive property is necessary (e.g., Evaluate
3x + 8 – 2x when x = 5. Evaluate 2(x – 1) when x = 8).
Scaffolding Activities for the Extended Indicator
Evaluate simple linear expressions for a specified value.
● Use a model to represent an unknown number in an expression. Present the expression
5+ and the model shown.
+
The expression may also be written with a variable, such as n. A variable is a letter that
stands for a number that is unknown.
+ n
When the value of the variable is given, the expression can be evaluated. For example,
5 + n can be evaluated when n = 5. Demonstrate substituting n with 5 tokens and evaluating
the expression 5 + n when n = 5. The value of the expression is 10.
+
Continue to demonstrate evaluating the expression 5 + n with other values for n.
Values may also be substituted when the expression has more than one step. Demonstrate
evaluating 6t + 7 when t = 3. Use a drawing or manipulatives as shown to demonstrate that
the expression can be interpreted as “6 groups of 3 added to 7.”
+
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 35MA 11.2.2 Algebraic Processes
A count of the triangles shows that the expression is equal to 25.
Be sure to emphasize that when a value is next to a variable, such as 6t, the operation
to use is multiplication. Therefore, the expression is 6 × t + 7. When 3 is substituted for t,
multiplication is evaluated first, so 6 × 3 = 18 and 18 + 7 = 25.
Continue to demonstrate evaluating two‑step expressions of the same type as 6t + 7 when
given the value of the variable. Demonstrate how to use manipulatives and create drawings
to evaluate the expression. Use other appropriate computation methods including, but not
limited to, using a calculator.
● Ask students to evaluate two‑step expressions such as 8y + 2 for y = 5 when given a model.
● Ask students to evaluate two‑step expressions such as 8 + h for h = 11 when not given a
model.
Evaluate linear expressions by combining like terms when given a specified value.
● Demonstrate that sometimes like terms can be combined to make an expression easier to
solve. For example, explain that in the expression 4n + 6 – 2n, the 4n and the 2n are like
terms because they use the same variable, n. The 2n can be taken away from 4n, since it is
subtraction, to make 2n. The new expression is 2n + 6. Therefore, if given the value n = 3,
then the expression could be rewritten as 2 × 3 + 6, which equals 12.
Continue to demonstrate how to evaluate other examples of linear expressions with like
terms, such as 3x + 4 + 2x + 6 when x = 4. First, combine the like terms, 3x and 2x, to get
5x. Then combine 4 and 6 to get 10. The expression may then be rewritten as 5x + 10.
Next, demonstrate evaluating the expression when x = 4. The new expression is 5 × 4 + 10,
which is 30. Use manipulatives and drawings as needed, as well as appropriate computation
methods including, but not limited to, using a calculator.
● Ask students to combine like terms in expressions such as 4h + 3h + 2 and 2v – 8 + 4v.
● Ask students to evaluate linear expressions by combining like terms.
4y + 5 + 2y – 1 when y = 3 4n + 3 + 5 when n = 5 5w + 8 – 3w when w = 2
Evaluate a linear expression using the distributive property.
● Explain that sometimes expressions include a number directly in front of a set of
parentheses, as in 5(x – 2). When the number is directly in front of a set of parentheses, it
means to multiply. Explain that to make the problem easier to solve, both terms inside the
parentheses can be multiplied by the number directly in front of the parentheses. In the
example 5(x – 2), both the x and the 2 are multiplied by 5. 5 × x = 5x and 5 × 2 = 10. The
new expression is 5x – 10.
Continue to demonstrate using the distributive property to simplify expressions such as the
following.
3(x + 4) 4(x – 5) 2(x + 6)
36 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.2.2 Algebraic Processes
● Demonstrate evaluating 4(x + 3) when x = 5 using the steps outlined below.
Step 1: Simplify the expression by multiplying both terms by 4.
4 × x = 4x and 4 × 3 = 12
The new expression is 4x + 12.
Step 2: Replace the x with the number 5.
4 × 5 = 20
The new expression is 20 + 12.
Step 3: Calculate 20 + 12 = 32.
Continue to demonstrate how to evaluate other linear expressions of the same type as
4(x + 3) when given a specified value for the variable. Use manipulatives and drawings as
needed, as well as appropriate computation methods including, but not limited to, using a
calculator.
● Ask students to simplify expressions such as the following using the distributive property.
5(x – 3) 2(x + 7) 4(x – 4)
● Ask students to evaluate linear expressions such as the following using the distributive
property.
7(f + 1) when f = 3 4(d – 3) when d = 4 5(y + 4) when y = 6
Prerequisite Extended Indicators
MAE 11.2.2.d—Add two linear expressions (e.g., (2x + 1) + (3x + 2) = 5x + 3).
MAE 7.2.2.c—Given the positive integer value of the single variable, evaluate an addition or
subtraction expression.
MAE 7.2.2.b—Identify equivalent expressions with one variable (2n + 3n is the same as 5n).
MAE 6.2.2.a—Identify whole number expressions using the distributive property (e.g., 2(3 + 4)).
Key Terms
addition, distributive property, expression, like terms, multiplication, subtraction, variable
Additional Resources or Links
https://www.map.mathshell.org/lessons.php?unit=7220&collection=8
https://www.engageny.org/resource/grade‑7‑mathematics‑module‑3‑topic‑lesson‑1
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 37MA 11.2.2 Algebraic Processes
MA 11.2.2.g
Solve linear and absolute value equations and inequalities.
Extended: Identify the absolute value of a negative integer.
Scaffolding Activities for the Extended Indicator
Use a horizontal number line to recognize the absolute value of an integer.
● Explain that the absolute value of a number is its distance from 0. Using the number line,
demonstrate the absolute value of –8 equals 8 because –8 is 8 units away from 0. Similarly,
demonstrate that the absolute value of 8 equals 8 because 8 is 8 units away from 0. Cut a
strip of paper that matches the distance from 0 to –8 on the number line. Place one end of
the strip of paper on the number line at –8 and the other end at 0. Then, move the strip of
paper on the number line so that one end is at 8 and the other end is at 0. Indicate that both
8 and –8 are the same distance from 0. Absolute value is indicated with two vertical lines;
|–8| = 8 can be read aloud as “the absolute value of negative eight equals eight.”
|–8| = 8 |8| = 8
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
38 Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High SchoolMA 11.2.2 Algebraic Processes
● Demonstrate recognizing number lines with a plotted point that has a given absolute value.
For example, present three number lines with points plotted at –14, –15, and –16 and
three questions: “Which number line has a point that has an absolute value of 14,” “Which
number line has a point that has an absolute value of 15,” and “Which number line has a
point that has an absolute value of 16?” Model answering each question by indicating the
correct number line.
0 0 0
–1 –1 –1
–2 –2 –2
–3 –3 –3
–4 –4 –4
–5 –5 –5
–6 –6 –6
–7 –7 –7
–8 –8 –8
–9 –9 –9
–10 –10 –10
–11 –11 –11
–12 –12 –12
–13 –13 –13
–14 –14 –14
–15 –15 –15
–16 –16 –16
–17 –17 –17
–18 –18 –18
–19 –19 –19
–20 –20 –20
Repeat the process using number lines showing points plotted at other negative integers,
such as three number lines with points plotted at –9, –10, and –11.
● Ask students to recognize number lines with plotted points that have a given absolute value.
Use a horizontal number line to identify the absolute value of an integer.
● Present a number line with a point plotted at –13. Explain that the absolute value of a
number is its distance from 0. Demonstrate finding the absolute value of –13 by pointing
to 0 on the number line and counting the number of units between 0 and –13. Since –13 is
13 units away from 0 on the number line, the absolute value of –13 equals 13.
–20 –19–18–17–16–15–14–13–12–11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
Continue demonstrating identifying the absolute value of other negative integers on a
number line. As appropriate, progress to identifying the absolute value of a negative integer
without using a number line.
● Ask students to identify the absolute value of negative integers when plotted on a number
line and without a number line.
Alternate Mathematics Instructional Supports for NSCAS Mathematics Extended Indicators—High School 39You can also read
