Standard 7-2: The student will demonstrate through the ...toolboxforteachers.s3.amazonaws.com/.../7th...Docs.pdf · relationships between fractions and fractional percents, mixed

$Page 1: Standard 7-2: The student will demonstrate through the ...toolboxforteachers.s3.amazonaws.com/.../7th...Docs.pdf · relationships between fractions and fractional percents, mixed$
Numbers and Operations Seventh Grade

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Indicator 7-2.1

Understand fractional percentages and percentages greater than one hundred.

Continuum of Knowledge

In the third grade, students worked with fractions that were less than or greater than one (3-2.5 and 3-2.6). In sixth grade, students are first introduced to the concept of percents (6-2.1).

In seventh grade, students’ knowledge is extended to percentages greater than one hundred (7-2.1).

Taxonomy Level

Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts

• Percent meaning per hundred Vocabulary

• % as a symbol Instructional Guidelines For this indicator, it is essential

• Recall that a fraction is part of a whole for students to:

• Recall and understand how to represent fractions and percentages less than 100

• Understand that with percent the whole is represented by 100%, no matter how many parts it includes.

• Represent percentages using concrete and/or pictorial models • Dividing one percent (1%) is the same as a fractional percent (a part of a

part). • Mixed numbers and improper fractions will always be a percent greater than

100.

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.


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For this indicator, it is not essential

for students to:

None noted Student Misconceptions/Errors

• It is difficult for students to understand how a percent can be greater than 100. Real world examples such in 2006, 4000 people answered yes to the survey question. In 2009, 10,000 people answered yes. Explain that 100% increase would be 8000 people but it increased over 100% because there’s more than 8000 people.

• Students who memorize rules for moving decimals may incorrectly move the decimal the wrong direction, or simply leave the decimal as it is and add a percent sign, or not understand decimal placement at all.

Instructional Resources and Strategies

• Show how to change fractions to percents by using equivalent fractions with 100 as the denominator (all examples should include factors of 100 as the denominator until 7-2.9 has been taught).

• Model changing decimals to percents by moving decimals to the right, modeling multiplication by 100, changing the place value by 100. Make explicit connection between these methods. They are all the different ways to do the same thing.

• Students should be exposed to concrete models to effectively connect the new learning to prior knowledge. Students Opportunities should also be provided that serve as a bridge between the knowledge of fractions less than/greater than one and fractional percents and percents that are greater than a whole, i.e. greater than 100%. Models should be used that include, but are not limited to, base ten blocks, 10 x 10 grid paper, and graph paper that will assist students in making the connection of fractions to percents less than one and mixed numbers to percents greater than 100. Model percentages like 1%, 6%, and 75%.

Assessment Guidelines The objective of this indicator is to understand, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To understand is to construct meaning. Conceptual knowledge is not bound by specific examples; therefore, the student’s conceptual knowledge of percents greater than 100 and less than one should include a variety of examples. The learning progression to understand requires students to recall the concept of a percent as a part of a whole and to understand the types of fractions and their meaning. Students use standard and nonstandard representations (7-1.8) to convey to support mathematical relationships between fractions and fractional percents, mixed number fractions and


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percents greater than 100 by exploring concrete and pictorial models. They interpret their use in the real world and generate meaningful concepts for fractional percents, for percents greater than one-hundred. They will also evaluate their real world situations and pose questions to prove or disprove their conjectures about percentages (7-1.2). Students should use

correct and clearly written or spoken words to communicate their understanding of these relationships (7-1.6).


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Indicator 7-2.2

Represent the location of rational numbers and square roots of perfect squares on a number line.


In the third grade students were first exposed to perfect square numbers while learning their basic multiplication facts such as 3 x 3, 6 x 6, etc. (3-2.7). In fifth grade, students compared whole numbers, decimals, and fractions (5-2.4). In sixth grade, students compared rational numbers and whole number percentages through (6-2.3).

In seventh grade, students represent the location of rational numbers and square roots of perfect squares on a number line (7-2.2).

In the eighth grade, students compare rational and irrational numbers (8-2.4).

Taxonomy Level


• <, >, <, >, = Vocabulary/Symbols

• rational number • square roots • perfect square

Instructional Guidelines For this indicator, it is essential

• Recall their knowledge of how to compare percents, fractions, decimals, and whole numbers

for students to:

• Recall the definition of rational numbers



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• Compute the square root of a number • Find equivalent fractions • Be able to justify their placement of a rational number on a number line • Understand the relationship to the numbers between which a given rational

number lies on a number line For this indicator, it is not essential

for students to:


• Students may order numbers on a number line with no regard to negative or positive signs on the numbers, incorrectly placing negatives to the right of zero and vice versa.

• Misunderstanding of fractions and mixed numbers values can lead to errors in placement of numbers on a number line.

• Not changing numbers to a common form before placing them on the number line is a common mistake. For example, not solving square roots before placement will lead to errors.


• Stress to students that it is easier to change all numbers involved to a common form before attempting to place them on a number line.

• Create a human number line with students placing rational numbers in various forms on a number line. They should all agree on the placement before putting it on the number line. Classmates will evaluate their placement for correctness.

• Students should be provided with opportunities to explore the location of fractions, decimals, percents, and square roots of perfect squares and to justify these placements of these representations as well as understand the relationship to the numbers between which a given value lies. Being able to support their placement with evidence will enable students to compare and order rational numbers, percents, and square roots of perfect squares using the order symbols.

Assessment Guidelines The objective of this indicator is to represent which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To understand is to construct meaning. To represent means to change from one representation to another; therefore, students construct conceptual understanding of rational numbers by changing them from numerical (number) to graphical form (number line). The learning progression to represent requires students to recall the concept of percents, fractions, and decimals as part of a whole and to make the connection to prior knowledge that the value of each part lies between two whole quantities.


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Students should explore how to represent these types of numbers in the correct location on a number line and justify

their placement of a rational number on a number line using general mathematical statements (7-1.5) based on inductive and deductive reasoning (7-1.3).


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Indicator 7-2.3

Compare rational numbers, percentages, and square roots of perfect squares by using the symbols <, >, <, >, and =.


Since kindergarten, students have used the terms more than, less than, and the same as to describe sets of objects (K-2.3). In first grade, students are introduced to the terms is greater than, is less than, and is equal to compare whole number quantities through 100 (1-2.5). In second grade, order symbols <, >, and = are used along with appropriate terms to compare whole numbers quantities through 999 (2-2.4), in third grade through 999,999 (3-2.1). In fourth grade, students compare fractions to benchmark fractions (4-2.9) and compare decimals through hundredths using terms less than, greater than, and equal to along with order symbols (4-2.7). In fifth grade students compare whole numbers, decimals, and fractions using order symbols (5-2.4). In the sixth grade students are first introduced to the concept of percents and learned to compare whole number percents through 100. They also learn the new symbols < and > (6-2.3).

In seventh grade, students compare rational numbers, percentages, and square roots of perfect squares by using the symbols <, >, <, >, and = (7-2.3). They also represent the location of rational numbers and square roots of perfect squares on a number line (7-2.2).

Taxonomy Level


• √ (square root) Vocabulary/Symbols

• x² (a number squared)



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• perfect square Instructional Guidelines For this indicator, it is essential

• Understand the meaning of rational numbers for students to:

• Find the square root • Understand the difference between ≤ and ≥ • Translate numbers to same form, where appropriate before comparing

numbers • Translate between the fraction and percents • Recall the benchmark fractions and common fraction – decimal equivalents.


for students to:


• Students may not understand that rational numbers include integers • Some students may not realize repeating decimals are rational numbers. • Students can also misread the order symbols and not include “or equal to”

for < and >. • Misunderstanding of comparisons of negative rational numbers. • Not converting improper fractions or finding square roots before comparing.


• Play a game of “Who Am I?” Students have a rational number written on a post-it note placed on their back upon entering the classroom. Students walk around asking yes/no questions to determine what their number is. Questions can include, “Am I greater than or equal to__?” Students will order themselves from left to right least to greatest in the front of the room, according to the responses they get to their questions.

• Provide true/false examples and give students the opportunity to justify their responses with evidence.

Assessment Guidelines The objective of this indicator is to compare, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. Conceptual knowledge is not bound by specific examples; therefore, the student’s conceptual knowledge should include numerous examples. The learning progression to compare requires students to recall common fraction/decimal equivalents and identify square roots. They understand equivalent symbolic expressions (7-1.4) and translate numbers to common form, if necessary. Students use their conceptual understanding to


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compare without dependent on a traditional algorithm and use concrete models to support understanding where appropriate. Students recognize mathematical symbols <, >, >, < and = and their meanings. As students analyze the relationships to compare percentages and rational numbers, they evaluate conjectures and explain and justify their answer to classmates and their teacher. Students should use

correct and clearly written or spoken words, variables and notation to communicate their reasoning (7-1.6).


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Indicator 7-2.4

Understand the meaning of absolute value.


In seventh grade, students understand the meaning of absolute value. The concept of absolute value is introduced for the first time in eighth grade.

In eighth grade, students apply the concept of absolute value (8-2.5).

Taxonomy Level


• absolute value symbol | | Vocabulary/Symbols


for students to:

• Understand that absolute value is a distance from zero, not a direction. • Understand that distance is always a positive value; therefore, the absolute

value is always positive For this indicator, it is not essential

for students to:

• Be able to solve problems involving absolute value. Student Misconceptions/Errors

• Students often have the misconception that distance can be a negative number when determining absolute value of a positive number, confusing the



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concept with opposite/additive inverse. The absolute value bars do not simply change the sign of the number inside the bars.

• Students may mistakenly use parentheses or brackets for the absolute value thinking that it doesn’t matter but these symbols do not mean the same thing.


• Everyday situations like positive/negative bank balances, above/below sea level, traveling directions, temperatures above/below zero should be included when developing an understanding of absolute value.

• Use visual imagery to demonstrate that moving away from zero, whether to the left or to the right, represents the idea of absolute value, a distance, not a direction.

• To deepen conceptual understanding, students need opportunities to compare absolute values using order symbols. For example, |-5 | -|4 |

• To deepen conceptual understand, explore the idea that taking the negative of an absolute value is the only way to get a negative value.

Assessment Guidelines The objective of this indicator is to understand, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To understand is to construct meaning; therefore, the student’s conceptual knowledge of absolute value should include numerous real world examples and non-examples. The learning progression to understand requires students to recall the concept of integers and their position in relation to zero. When directed to determine the absolute value of a number, students should understand that absolute value is a distance away from zero and does not include which direction (positive or negative) away from zero. They illustrate this understanding by representing absolute value relationship on the number line. Students use

their understanding to generate and solve complex problems to deepen conceptual knowledge. They use correct and clearly spoken words, variables and notation to communicate their understanding (7-1.6).


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Indicator 7-2.5 Apply ratios, rates, and proportions to discounts, taxes, tips, interest, unit costs and similar shapes Continuum of Knowledge In sixth grade, students understand the relationship between ratio/rate and multiplication/division (6-2.6). In seventh grade, students apply ratios, rates, and proportions to discounts, taxes, tips, interest, unit costs and similar shapes (7-2.5). In eighth grade, students apply ratios, rates and proportions. Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts

• Ratios • Rates • Discount • Taxes • Tips • Interest • Unit Cost • Similar Shapes


for students to:

• Understand the meaning of ratio, rate and proportions • Develop strategies for determining discounts


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• Connect the similar ideas computing discounts, taxes, tips and interest in order to develop strategies

• Understand that with discounts, they pay less and with taxes, tips and interest they pay more

• Understand the concept of percents • Multiply by a decimal • Understand the relationship between unit cost and ratio and rate • Understand the proportionality of similar shapes • Solve proportions using an appropriate strategy • Explore real world examples of discounts, taxes, tips and interest • Determine the final cost after a discount, taxes and interest are applied


for students to:

• Apply ratios, rates and proportions for problems beyond those situations outlined in the indicator

Student Misconceptions/Errors Students may still struggle with converting decimals. Instructional Resources and Strategies Although students are not required to compute the final cost, it may helpful for them to understand that discounts (subtracting from original amount) and taxes, tips and interest (add to the original amount). Representing these concepts as pictorial or concrete models may help students deepen their understand these procedures. For example, using a hundreds charts to represent $100 dollars. Shade appropriate blocks to represent discounts like 20% off mean subtracting 20 blocks or interest of 30% means adding 30 more blocks. The cross multiplying algorithm for solving proportions is a valid strategy that should be taught with understanding. Using proportional reasoning is another powerful strategy that focuses more on a conceptual understanding of proportional relationship as opposed to a traditional algorithm. Proportional reasoning always allows students make estimates. Assessment Guidelines The objective of this indicator is to apply which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Although the focus of the indicator is procedural, students will need a conceptual understand of the concepts of discount, taxes, tips and interest. The learning progression to apply requires students to recall and understand the meaning of the concepts of discounts, taxes, tips,


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interest, unit cost and similar shapes. Students explore a variety of problems in context to generalize connections (7-1.7) between these concepts, appropriate computational procedures and these real world situations (7-1.7). They analyze pictorial and/or concrete models to gain a conceptual understanding of these procedures. They generate mathematical statements (7-1.5) related to how to use ratios, rates and proportions to solve problems and use

correct and clearly written and spoken words, variable and notation to communicate their understanding (7-1.6). Students should engage in repeated practice to gain fluency in these procedures.


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Indicator 7-2.6

Translate between standard form and exponential form.


In sixth grade, students applied strategies and procedures to determine values of powers of 10 up to 106

In seventh grade, students translate between standard form and exponential form.

(6-2.7) and represented whole numbers in exponential form (6-2.9).

Taxonomy Level

Cognitive Dimension: Understand Knowledge Dimension: Procedural Key Concepts

• standard form Vocabulary/Symbols

• exponential form • base • exponent • squared • cubed • prime factorization


for students to:

• Understand the structure of exponential form • Recall how to apply the procedure of prime factorization • Understand that any number to the power of one is the number • Understand that any number to the zero power is one except 00



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for students to:

• Perform operations with numbers in exponential form Student Misconceptions/Errors

• Students multiply the base number by the exponent. For example, 23

• Misinterpreting the terms squared or cubed.

= 2 x 2 x 2 not 2 x 3.

• Students may multiply the base by the exponent. Ex. 23

• Students may not understand the terminology, such as “squared” or “cubed”. = 2 x 3 = 6.

• Student may incorrectly think that a base with an exponent of zero equals zero. (such as 40 = 0). The student may incorrectly think every number raised to the zero power is zero forgetting that there is the exception of 0

0


• A component of this indicator is very similar to students working with prime factorization.

One strategy for explaining why any number to the zero power is one is through the use of patterns. For example, at each stage in the pattern, the exponent decreases by one, the answer is divided by 5. Continue the pattern and eventually 50

5 = 1.

4

5 = 625

3

5 = 125

2

5 = 25

1

5 = ?

0

= ?

The same process can be done with any base. For example, start with 24

. As the exponent decreases by one, the answer is divided by 2.

Assessment Guidelines The objective of this indicator is to translate, which is in the “understand procedural” knowledge cell of the Revised Bloom’s Taxonomy. To understand a procedural means that students not only know how to translate but they also develop the conceptual relationship between standard form and exponential form using a variety of examples. The learning progression to translate requires students to recognize the standard and exponential form of a number. Students explore patterns to generalize mathematical statements about any number to the power of one. They explain and justify their answers using correct and clearly written or spoken words and notation (7-1.6).


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Indicator 7-2.7

Translate between standard form and scientific notation.


In sixth grade, students applied strategies and procedures to determine values of powers of 10 up to 106

In seventh grade, students translate between standard form to exponential form and to scientific notation (7-2.7). They also translate between standard form and scientific notation (7-2.6). This is the first time students transfer numbers between standard form and exponential form and scientific notation.

(6-2.7) and they represented whole numbers in exponential form (6-2.9).

Taxonomy Level


• standard form Vocabulary/Symbols

• exponential form • scientific notation • base number


• Understand and expand powers of 10 for students to:

• Understand the purpose of scientific notation • Understand the structure of scientific notation • Determine if a number is written in scientific notation • Recognize small and large numbers



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• Connect the sign of the exponent to the direction in which the decimal is moved

• Convert from standard to scientific and vice-versa For this indicator, it is not essential

for students to:

• Perform operations with numbers in scientific notation

Student Misconceptions/Errors

• Students may be confused by which direction they should move the decimal point when translating from one form to another/


• Students should be provided with a variety of examples, both very large and very small, as well as decimal and whole numbers.

Assessment Guidelines The objective of this indicator is to translate, which is in the “understand procedural” knowledge cell of the Revised Taxonomy. To understand a procedural means that students not only know how translate but they also develop the conceptual relationship between standard form and scientific notation using a variety of examples. The learning progression to translate requires students to recognize the standard form and scientific notation form of a number. Students need to explore and connect numbers that are very large or very small to meaningful points of reference to understand their magnitude in the real world. They analyze numbers to determine the number part and the power of 10. They explain and justify their answers to their classmates and teacher using

correct and clearly written or spoken words, variables and notation to communicate their understanding (7-1.6).


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Indicator 7-2.8

Generate strategies to add, subtract, multiply, and divide integers.


In sixth grade, students develop a meaning of integers (6-2.2).

In seventh grade, students generate strategies to add, subtract, multiply and divide integers (7-2.8).

In eighth grade, students apply an algorithm to add, subtract, multiply and divide integers (8-2.1).

Taxonomy Level

Cognitive Dimension: Create Knowledge Dimension: Conceptual Key Concepts

• Integer • Whole Number • Positive • Negative • Operations


for students to:

• Understand the meaning of integers • Understand how to represent integers • Explore concrete models, manipulatives, and pictorial representations


• Perform symbolic operations with integers for students to:



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• Students may misunderstand zero pairs. • When using a number line to explore operations, students could misinterpret

direction on a number line.


• Review with students situations with negative numbers they may encounter Assessment Guidelines The objective of this indicator is to generate which is in the “conceptual knowledge” of the Revised Taxonomy. To create is to put together elements to form a new, coherent whole or to make an original product. Conceptual knowledge is not bound by specific examples. The learning progression to generate requires students to recall and understand the concept of integers. Students explore problem situations (story problems) and explore various strategies to solve those problems by applying their conceptual knowledge of integers. Students translate their understanding of concrete and/or pictorial representations by generalizing connections between their models and real world situations (7-1.7). Students should use these procedures in context as opposed to only rote computational exercises and use correct and clearly written or spoken words to communicate about these significant mathematical tasks (7-1.6). Students formulate questions to prove or disprove their methods (7-1.2) and generate mathematical statements (7-1.5) about these operations. They should evaluate

the reasonableness of their answers using appropriate estimation strategies.


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Indicator 7-2.9

Apply an algorithm to multiply and divide fractions and decimals.


In sixth grade, students generated strategies to multiply and divide fractions and decimals. Sixth was the first time to the concept of multiplying and dividing fractions and decimals was introduced (6-2.5).

In seventh grade, students apply an algorithm to multiply and divide fractions and decimals (with numbers only).

In eighth grade, students understand the effect of multiplying and dividing rational numbers by another rational number (8-2.2).

Taxonomy Level

Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts

• Denominator Vocabulary/Symbols

• numerator • Quotient • Product


• Understand that the product of fractions and decimals does not always result in a larger number

for students to:

• Understand the meaning and concept of fractions and decimals



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• Connect their generated strategies and their concrete/pictorial models with their algorithm

• Understand that multiplication does not always result in a larger answer and division does not result in a smaller answer

• Gain computational fluency • Estimate products and quotients of fractions and decimals to determine

reasonableness of answers once a problem is solved.


for students to:

None noted


None noted


• Although the focus is on computational fluency, students will need opportunities to connect prior experiences using concrete and pictorial experiences to the symbolic operations they are using now.

• Using real world problems to explore algorithms gives the students a context in which to use these processes.

• Using the distributive property may be especially useful with students who understand that 2 ½ means 2 + ½ and can multiply ¾ by 2 ½ as ¾ times 2 and ¾ times ½.

Assessment Guidelines The objective of this indicator is apply, which is in the “apply procedural” of the Revised Taxonomy. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem or problem situation. Although the focus is to gain computational fluency with addition and subtraction of fractions, the learning progression should integrate strategies to enhance both conceptual and procedural knowledge. The learning progression to apply requires students to understand decimals and fractional forms such as mixed numbers, proper fractions, and improper fractions. Students should apply their conceptual knowledge of fractions and decimals to transfer their understanding of concrete and/or pictorial representations to symbolic representations (numbers only) by generalizing connections among a variety of representational forms and real world situations (6-1.7). Students use these procedures in context as opposed to only rote computational exercises and use correct and clearly written or spoken words to communicate about these significant mathematical tasks (6-1.6). Students engage in repeated practice using pictorial models, if needed, to support learning. Lastly, students should evaluate

the reasonableness of their answers using appropriate estimation strategies.


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Indicator 7-2.10

Understand the inverse relationship between squaring and finding the square roots of perfect squares.


In the third grade students had their first experiences with perfect squares as they learned basic multiplication facts such as 4 x 4 and 7 x 7 (3-2.7). Students developed understanding of the inverse relationship between addition and subtraction in first grade (1-2.7) and multiplication and division in third grade (3-2.8).

In seventh grade, students develop an understanding of the inverse relationship between squaring and finding square roots of perfect squares (7-2.10).

In the eighth grade, students will begin applying strategies and procedures to approximate the square roots or cube roots of numbers less than 1,000 between two whole numbers (8-2.6). Cube roots will be a new topic in eighth grade but requires an understanding of square roots.

Taxonomy Level


• √ (square root) Vocabulary/Symbols

• x² (a number squared) • perfect square • inverse relationship



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• Understand the concept of squaring for students to:

• Understand the concept of square root • Use concrete/pictorial models to connect the concept of a squared number

and a corresponding geometric square. For example, finding the length of a side of a square with area equal to 25 units and finding the square root of 25 are basically the same question.


for students to:

• Know whether or not the square root can be positive or negative. For example, the square of 25 can be -5 or +5.


• Students may have the misconception that square root means to divide by 2. Instructional Resources and Strategies

• Using concrete and pictorial models can help students build conceptual understand of this inverse relationship. One strategy involves asking students to explore finding rectangles that have an area of 36 and use grid paper to shade in the various shapes they identify. They should recall that area is calculated by multiplying the length and the width of the figure. Generalize a statement describing the area and the side lengths of the figure that forms a perfect square.

• Learning opportunities should include both models and numerical examples. Assessment Guidelines The objective of this indicator is to understand, which is in the “understand conceptual” knowledge cell of the Revised Bloom’s Taxonomy. To understand is to construct meaning. Conceptual knowledge is not bound by specific examples; therefore, the student’s conceptual knowledge should include a variety of examples. The learning progression to understand requires students to recall and understand the meaning of inverse relationships, squaring and square roots. Students also recall what is unique about the side lengths of squares. Students explore the relationship between area and square roots using inductive and deductive reasoning (7-1.3). They generalize

this connection using correct and clearly written or spoken words (7-1.6)to demonstrate their understanding of the inverse relationship.

Algebra Seventh Grade

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Standard 7-3: The student will demonstrate through the mathematical processes an understanding of proportional relationships. Indicator 7-3.1 Analyze geometric patterns and pattern relationships. Continuum of Knowledge In third grade, students created numeric pattern that involve whole-number operations (3-3.1). In fourth grade, analyzed numeric, nonnumeric, and repeating patterns involving all operations and decimal patterns through hundredths (4-3.1). In fifth grade, students represented numeric, algebraic and geometric pattern in words, symbols, algebraic expressions and algebraic equations (5-3.1). In sixth grade, they analyzed numeric and algebraic pattern and pattern relationships (6-3.1). In seventh grade, students analyze geometric pattern and pattern relationships (7-3.1). In eighth grade, students represent algebraic relationship with equations and inequalities (8-3.2). Taxonomy Level Cognitive Dimension: Analyze Knowledge Dimension: Conceptual Key Concepts

• Geometric patterns Vocabulary

• Triangular numbers • Square numbers


for students to:

• Understand the meaning of geometric pattern • Compare the magnitude of numbers • Understand the concept of growing patterns


for students to:

• None noted


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Student Misconceptions/Errors Students are accustomed to looking for repeating or growing patterns that involve simple addition – either perform the operation of addition to get the next step or add another shape/figure, etc. When observing the class, look for students who may not have a solid understanding of those simple relationships and are thus not able to move to more complex thinking. Instructional Resources and Strategies

• Some texts define geometric sequences and “figurate numbers” differently. Because the focus of this indicator is geometric patterns, we will make no distinction other than to explain how texts see them differently. The explanation is provided to alert you as a teacher that student learning opportunities should include all types without

students having to identify the various types. Also, some texts label geometric patterns as growing patterns.

• Geometric sequence – Each successive term is determined by multiplying the preceding term by a fixed number, the ratio. Figurate numbers

– typically we think of triangular or square numbers – representations can be arranged in the shape of triangles or squares.

• Help students understand that just as in earlier grades, it is not always

possible to draw out an entire pattern to determine the number of items/shapes that will be in say, the 100th

term. Therefore, it is critical to begin to look at how the steps fit together or are related to form the next step.

Assessment Guidelines The objective of this indicator is analyze, which is in the “understand conceptual” knowledge of the Revised Bloom’s Taxonomy. To understand is to construct meaning. Conceptual knowledge is not bound by specific examples; therefore, the student’s conceptual knowledge of geometric patterns and pattern relationships should include a variety of examples. The learning progression to analyze requires student to recall and understand the meaning of geometric patterns. Students distinguish between relevant and irrelevant numbers in mathematical word problems, compare the magnitude of numbers and compare the relationship among objects in order to generate a mathematical statement (7-1.5) that can be used to predict the next element in the pattern. Students use

correct and clearly written or spoken words, variables, and notations to communicate their understanding (7-1.6).


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Standard 7-3: The student will demonstrate through the mathematical processes an understanding of proportional relationships. Indicator 7-3.2 Analyze tables and graphs to describe the rate of change between and among quantities. Continuum of Knowledge In fifth grade, students analyzed change over time (5-3.2).

In seventh grade, students analyze tables and graphs to describe rate of change and understand slope as a constant rate of change (7-3.2).

In eighth grade, students identify the slope of a linear equation from a graph, equation and/or table (8-3.7)

Taxonomy Level Cognitive Dimension: Analyze Knowledge Dimension: Conceptual knowledge Key Concepts

• Rate of change Vocabulary

• Slope • Independent • Dependent


for students to:

• Examine patterns in tables and graphs • Describe the change among quantities • Determine if the rate of change is constant or not. • Represent rate of change as “vertical change divided by horizontal change” • Represent the rate of change as “change in the dependent variable divided

by change in the independent variable” • Discover that the rate of change and slope are one in the same.


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• Calculate slope using a formula

for students to:


A common mistake that students make is to put the horizontal (run) quantity over the vertical (rise) quantity when computing rate of change. This is an indication that they lack conceptual understanding.


• The focus of the indicator is on students building a conceptual understanding of rate of change. This conceptual understanding is foundational for their future studies in Algebra I and is a major concept in Calculus.

• Navigating through Algebra Grades 6-8 “Stories to Graphs and Graph to Stories”: Students translate verbal description of distance-time relationship to graph form and vice versa. An emphasis is placed on student explaining why is happening during a certain interval of time.

• http://www.analyzemath.com/Slope/Slope.html Assessment Guidelines The objective of this indicator is to analyze which is in the “analyze conceptual” knowledge cell of the Revised Taxonomy. To analyze means to break down materials into parts (vertical change and horizontal change) and determine how the parts related to one another and the overall structure (graph or table). The learning progression to analyze requires students to understand the meaning of the terms independent and dependent variable and find each from a table or a graph. As students explore real world data, they determine the change for each and use their prior knowledge of rate to create a verbal description of the observed changes. They then translate their description to numerical form indicating appropriate units. Students should evaluate their answers and pose questions to prove or disprove their reasoning (7-1.2). Students generalize mathematical statements about rate of change and use

that understand to solve other problems that model physical, social and mathematical phenomena (7-1.1).

http://www.analyzemath.com/Slope/Slope.html�


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Standard 7-3: The student will demonstrate through the mathematical processes an understanding of proportional relationships. Indicator 7-3.3 Understand slope as a constant rate of change. Continuum of Knowledge In fifth grade, students analyzed change over time (5-3.2).

In seventh grade, students analyze tables and graphs to describe rate of change and understand slope as a constant rate of change (7-3.2).

In eighth grade, students identify the slope of a linear equation from a graph, equation and/or table (8-3.7)

Taxonomy Level Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts

Vocabulary

• Slope • Rate of change • Independent variable • Dependent variable • Horizontal change • Vertical change • Steepness


for students to:

• Understand unit rates • Plot points • Generate a table of values • Recognize a pattern of constant change • Know the relationship between rate of change and the steepness of the line • Make connections that an increasing line is a positive slope and a decreasing

line is a negative slope • The relationship between the rate of the change of the table and the rise/run


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of the graph • Understand that the graph of the line has a constant rate of change


for students to:

• Use a formula to compute slope • Compute the slope from a line using rise over run triangles


• Students may misunderstand that change must be constant• Students mistakenly write slope as “change in x divided by change in y”

.


• Having an understanding of the multiple representations is essential to students building conceptual understanding of slope. Students should examine changes in the table of values and the graph simultaneously to understand that both representations should convey the same meaning. This creates that connection between the steepness of the line and the rate because as the rate changes in the table, the steepness changes as well.

• A tool such as a CBR (Calculator Based Ranger by Texas Instrument) is a great way to generate a lot of interest and make learning about slope fun experience for students. Activities for the CBR at the middle level can be found at: http://education.ti.com/educationportal/sites/US/nonProductSingle/activitybook_cbr_cbl_73_datacollection.html

Assessment Guidelines The objective of this indicator is to understand which is in the understand conceptual knowledge cell of the Revised Taxonomy. To understand is to construct meaning; therefore, students will construct meaning related to the relationship between the steepness of a line and the rate of change. The learning progression to understand requires students to understand the meaning of rate of change and understand a process for determining the rate of change. Students explore the relationship between the changes in the independent/dependent variables in a table of values and changes in the steepness of a line by examining these representations simultaneously. Students generalize connections (7-1.7) between these forms and generate

mathematical statements (7-1.5) to summarize these connections using correct and clearly written or spoken words (7-1.6)

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Standard 7-3: The student will demonstrate through the mathematical processes an understanding of proportional relationships. Indicator 7-3.4 Use inverse operations to solve two-step equations and two-step inequalities. Continuum of Knowledge In sixth grade, students use inverse operations to solve one-step equations that involve only whole numbers (6-3.5). They have not yet had any instruction in solving inequalities. The foundation begins to be built in the sixth grade with simple one step equations (with whole numbers only). In seventh grade, students use inverse operations to solve two-step equations and two-step inequalities (7-3.4). In eighth grade, students apply a procedure to solve multi-step equations (8-3.4). In Algebra I, students carry out a procedure to solve linear equations for one variable algebraically (EA-4.7) Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts

• Equation Vocabulary

• Expression • Inequality • Inverse • Inverse operations • Multiplicative inverse • Additive inverse • Two-step equations • Two-step inequalities • Solution


for students to:

• Understand the concept of inverse operations • Understand the concepts of equivalency and inequality


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• Use substitution to check solutions of two-step equations and inequalities • Distinguish between the meaning of < verses ≤ and > verses ≥, particularly

in regards to their graphs • Use concrete and pictorial models to solve equations


for students to:

• Solve equations or inequalities that require operational use with negative integers


• Exercise caution when introducing solving inequalities that include negative numbers since students to not use symbolic integer operation until the eighth grade. The tendency is to simply tell students to reverse the inequality symbol when multiplying or dividing by a negative number. However, without understanding why, a student will soon forget that “rule” of inequalities. It is important that students understand "why" the sign reverses when multiplying or dividing by a negative number.

• Student may perform inverse operation to only one side. • Student may use incorrect inverse operation. • Students may become confused when given problems with the variable on

the right side of the equals sign such as 25 = 3x + 7. • Some students may try to simplify across the equal sign. An example may

be in x + 7 = 7, the student may add 7 to the right side to get x = 14. Instructional Resources and Strategies

• To solve an equation means to find value (s) for the variable that make the equation true. Pan balance may be used to develop skills in solving equations with one variable. “The balance makes it reasonably clear to students that if you add or subtract a value from one side, you must add or subtract a like value from the other side to keep the scales balanced. Students can generalize this idea and apply it to inequalities. What would the inequality look like on a balance scale? Unlike with linear equations where only one value makes the scale balanced, linear inequalities can have several values that make it unbalanced.

• Students may undo the multiplication or division first then undo the addition or subtraction. Although, this process is not incorrect, students don’t realize that they have to divide every term by the number.

• Having students find errors or compare strategies uses different cognitive processes than when students simply apply a procedure. An example:


9

Ask students to identify and correct the error in the following solution of the

equation 3 x - 7 = 22.

3 x – 7 = 21 3 x – 7 =

3 3 21

x – 7 = 7 x = 14

Other questions may include

• What does it mean to “isolate the variable”?

• How do you check an equation to see if your solution is correct?

• Why do you apply the order of operations in reverse when solving equations?

Assessment Guidelines The objective of this indicator is use which is in the “apply procedural” of the Revised Taxonomy. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem or problem situation. Although the focus is to gain computational fluency with solving two step equations and inequalities, the learning progression should integrate strategies to enhance both conceptual and procedural knowledge. The learning progression to use requires students to explore the concepts of equivalency, inequality and variables using concrete models such as balance scales. Student use this understanding of balance to analyze a variety of examples of simple two step equations and inequalities. Students use inductive reasoning (6-1.3) to generalize connections (6-1.7) among types of equations/inequalities and generate mathematical statements (6-1.5) related to how these equations can be solved. They use concrete manipulatives and pictorial to support their conceptual understanding. Student engage in meaningful practice to support retention of these processed and check

their answers.


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Standard 7-3: The student will demonstrate through the mathematical processes an understanding of proportional relationships. Indicator 7-3.5 Represent on a number line the solution of a two-step inequality. Continuum of Knowledge In sixth grade, students are introduced to the concept of equality and inequality (6-3.3). In seventh grade, students represent on a number line the solution of a two-step inequality (7-3.5) and solve two step equations and inequalities (7-3.4). In eighth grade, students solve inequalities and multi-step equations and represent their solutions (8-3.2). Taxonomy Level Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts

Open circle <, > Vocabulary

Closed circle ≤, ≥ Instructional Guidelines For this indicator, it is essential

for students to:

• Distinguish between the meaning of < verses ≤ and > verses ≥ • Understand the inequalities have several solutions unlike equations that only

have one • Understand the meaning of open and closed circles • Understand how to determine the direction in which to shade • Translate from graph to inequality and inequality to graph • Check values from the graph by using substitution


for students to:

• Determine solutions and represent on a number line the solutions to multi-step inequalities


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Student Misconceptions/Errors Students may solve inequalities but fail to graph solutions correctly because they are unclear about in which direction to shade. This is an issue especially with problems where the variable is on the right side. For example, -5 < x. Instructional Resources and Strategies

• This strategy will introduce students to “open” and “closed” notations when graphing solutions. Have students write “open” and “closed” at the top of their notes and place the appropriate signs beneath each. Draw examples of graphs for students. Remind students also that the number line goes on indefinitely in each direction. Remember to question students to name several integers that will make the solution true. For example, if x < 5 is the solution, students may substitute 4, 0, or -3 into the inequality to prove the solution true. Have students work in cooperative pairs to solve and graph the solutions to two step inequalities. They must agree on a solution before moving to the next problem.

• Sketch several graphs on the overhead that show the solution sets of several

inequalities. Have students state the corresponding inequalities.

• Explain to your partner the difference in using an open and a closed circle when graphing the solution of an inequality.

Assessment Guidelines The objective of this indicator is to represent which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To represent is to translate from one form to another and to understand is to construct meaning; therefore, students are constructing meaning between the graphical representation and algebraic representation of solution to inequalities. The learning progression to represent requires students to understand the meaning of inequality symbols and open/closed circles. Students understand that the solution set to an inequality consists of multiple solutions. They explore this concept through the use of concrete models such as a balance scale. They use this understanding to graph the given inequality on a number line. They recognize that the graph and the inequality are equivalent symbolic representations of the same relationship (7-1.4). Students then evaluate their graph to determine if it is correct by verifying that solutions are in the shaded region. Students explain and justify

their answers using correct and clearly written and spoken words (7-1.6).


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Standard 7-3: The student will demonstrate through the mathematical processes an understanding of proportional relationships. Indicator 7-3.6 Represent proportional relationships with graphs, tables, and equations. Continuum of Knowledge In sixth grade, students used proportions to determine unit rates (6-5.6). In seventh grade, students represent proportional relationships with graphs, tables, and equations. In eighth grade, students translate among verbal, graphic, tabular and algebraic representations of linear funtions (8-3.1). Taxonomy Level Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts

• Proportional relationships Vocabulary

• Directly proportional • Inversely proportional • Non proportional • Multiple representations


for students to:

• Represent directly proportional relationships • Identify the unit rate as the slope of the related line • Represent inversely proportional relationships • Recognize if a relationship is directly, inversely or not proportional • Understand the relationship between the multiple representations (graph,

table and equation) of proportional relationships For this indicator, it is not essential

for students to:

• Represent non-proportional relationships


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Student Misconceptions/Errors A common misconception is to think two quantities are directly proportional simply because they increase at a constant rate. In fact, it is the ratio of the quantities that must remain constant. Instructional Resources and Strategies

• An emphasis should be placed on the multiple representations of the concept of proportionality. Student should explore the representations (graph, table and equation) simultaneously and discuss how each shows the same relationship only in a different form

• Give student opportunities to explore relationships that are not either directly or inversely proportional – even if it at first it appears they fit the definition. For example:

Example A: You are 14 and your sister is half your age – 7. Next year you will be 15 and your sister will be 8 – more than half your age. While both ages increase, the factor does not remain constant. Thus, there is no direct proportionality. (NOTE: It would be best to state the first sentence of this example and have student make a T-Chart and a graph of the ages for the next six years. Remind students to label and title “Ages”. Next ask them how this example is like and different from the other T-Charts and graphs. Both factors increase but not by a constant factor. Also, you may want to ask students which definition does this appear to fit at first glance and why does it not meet the requirements?) Note: These examples were adapted from Algebra To Go

, Great Source Publishing, 2000.

• Ask students to provide examples of each of the three types of relationships and to justify their reasoning.

• Use a ticket out check method - Show students a graph (table) depicting

directly proportional, inversely proportional or non-proportional relationships. Ask students to select a graph, identify the relationship and justify their identification. Have students turn in their response as a ticket out of the class.

• Use either of the above mentioned strategies but substitute a chart or table

for the graph.

Assessment Guidelines The objective of this indicator is to represent which is in the understand conceptual knowledge cell of the Revised Taxonomy. To represent means to translate from one form to another and to understand is to construct meaning; therefore, students


14

construct meaning for proportional relationships by translating among graph, tables, and equations. The learning progression to represent requires students to recall and understand the meaning of proportional relationships. Students analyze teacher generated examples and generalize mathematical statements about proportionality. They use deductive reasoning to move from general statements to specific statements about directly proportional and inversely proportional relationships. They evaluate their conjectures (7-1.2) by applying their understanding to other examples and explain and justify their answers to their classmates and teacher. Students recognize and understand that the representations (graphs, tables, and equations) are equivalent symbolic representations of the same relationship (7-1.4). They progress to generating

and solving more complex problem (7-1.1).


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Standard: 7-3: The student will demonstrate through the mathematical processes an understanding of proportional relationships. Indicator 7-3.7 Classify relationships as directly proportional, inversely proportional, or non-proportional. Continuum of Knowledge In sixth grade, students used proportions to determine unit rates (6-5.6). In seventh grade, students represent proportional relationships with graphs, tables, and equations. In eighth grade, students translate among verbal, graphic, tabular and algebraic representations of linear funtions (8-3.1). Taxonomy Level Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts

• Directly proportional Vocabulary

• Inversely proportional • Non-proportional • Multiple representations (table, graph and equation)


• Understand the meaning directly proportional for students to know:

• Recognize examples of • Inversely Proportional • Non-proportional – A relationship exists between quantities but the

relationship is neither directly proportional nor inversely proportional. For this indicator, it is not essential

• None noted for students to:


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Student Misconceptions/Errors A common misconception is to think two quantities are directly proportional simply because they increase at a constant rate. In fact, it is the ratio of the quantities that must remain constant. Instructional Resources and Strategies

• One definition of inversely proportional states that as one quantity increases,

the other quantity decreases by the same constant factor. Note: Some definitions may say the quantities change by reciprocal factors. That definition is correct also because for one quantity to decrease proportionally, the reciprocal factor must be applied.

• When using two order pairs (x1, y1) and (x2, y2

Directly proportional is

), teachers may refer to these alternative forms:

1

1

xy

= 2

2

xy

. Inversely proportional is 1

2

2

1

yy

xx

=.

• An overall curriculum theme seen throughout the seventh grade Algebra strand is the concept of a constant rate of change (slope) and the tables, equations, and graphs that result from a relationship that has a constant rate of change. It is important that students be given ample opportunities to discover the connection between direct proportionality and the table, equation, and graph this relationship produces as it leads to the gentle introduction of slope and then linear functions in the eighth grade. As students graph directly proportional relationships, they should be able to identify the unit rate as the slope of the related line.

• Representing inversely proportional relationships in tables, equations, and graphs allows students to understand that not all tables and equations produce similar graphs and that slope only exists when there is a "constant" rate of change. This understanding is important as the focus in eighth grade will be on the table, equations, and graphs derived from linear functions.

Assessment Guidelines The objective of this indicator is to classify which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To classify means to determine whether something belongs to a category. The learning progression to classify requires students to


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The objective of this indicator is to classify which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To understand is to construct meaning and conceptual knowledge is not bound by specific examples; therefore, students should gain a conceptual understanding of proportionality in order to classify any shape. The learning progression to classify requires students to understand the characteristics of directly proportional, inversely proportional and non-proportional relationship represented as graphs, tables and equations. Students analyze these representations and compare them against known characteristics. As students analyze these relationships, they categorize them as directly, inversely or non-proportional. Students explain and justify

their answers using correct and clearly written and spoken words, variables and notation to communicate their understanding (7-1.6).

Geometry Seventh Grade

1

Indicator 7-4.1

Analyze geometric properties and the relationships among the properties of triangles, congruency, similarity, and transformations to make deductive arguments.


In fifth grade, students classified shapes as congruent (5-4.3). In sixth grade, they identify the transformations used to move a polygon from one location to another in the coordinate plane (6-4.5) and classified shapes as similar (6-4.8).

In seventh grade, students analyze geometric properties and the relationships among the properties of triangles, congruency, similarity, and transformations to make deductive arguments (7-4.1).

In eighth grade, students apply a dilation to a square, rectangle or right triangle in a coordinate plane (8-4.3).

Taxonomy Level

Cognitive Dimension: Analyze Knowledge Dimension: Conceptual Key Concepts

Vocabulary

• Congruency • Similarity • Transformations • Ratios • Deductive • Conjectures • Proportional

Instructional Guidelines For this indicator, it is essential for students to:

Standard 7-4: The student will demonstrate through the mathematical processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments, the results of the intersection of geometric shapes in a plane, and the relationships among angles formed when a transversal intersects two parallel lines.


2

• Recall the properties of polygons • Understand the properties of congruency. • Understand the properties of similarity. • Perform transformations (reflection, rotation, translation) • Use deductive reasoning to move from general to specific statements about

triangles, congruence, similarity and transformation For this indicator, it is not essential

for students to:

• Prove congruency and similarity using formal methods of proofs. • Find measures of missing sides and angles of triangles based on congruency

and similarity. Student Misconceptions/Errors

• Students may sometimes get confused differentiating between congruency and similarity. It is very important for students to be given the opportunity to investigate the similarities and differences between similar and congruent triangles.


• The wording of the indicator implies that students make deductive arguments about the concepts of congruency, similarity and transformations using polygons other than triangles. Students analyze triangles, as well, to make deductive agruments about their properties and relationships. A primary focus is for students to begin to think more formally about the concepts of congruency, similarity and transformations and they how might be used when proving relationships.

• Congruency http://nlvm.usu.edu/en/nav/frames_asid_165_g_3_t_3.html?open=instructions&from=category_g_3_t_3.html

• Congruency, Similarity, and Symmetry http://standards.nctm.org/document/eexamples/chap6/6.4/part3.htm

Assessment Guidelines The objective of this indicator is to analyze, which is in the “analyze conceptual” knowledge cell of the Revised Taxonomy. Conceptual knowledge is not bound by specific examples; therefore, the student’s conceptual knowledge of similarity, congruency, transformation and triangles should include exploring a variety of polygons. The learning progression to analyze requires students to compare congruent and similar polygons to understand their similarities and differences. Students use inductive and deductive reasoning to generalize connections among these shapes. They generalize mathematical statements (7-1.5) about these

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connections and pose follow-up questions to prove or disprove their conjectures (7-1.2). They explain and justify their deductive arguments using correct and clearly written or spoken words or notation to communicate

(7-1.6) their findings to their

peers and teacher.


4


Indicator 7-4.2 Explain the results of the intersection of two or more geometric shapes in a plane Continuum of Knowledge In third grade, students classified lines and line segments as either parallel, perpendicular, or intersecting (3-4.3). In seventh grade, students explain the results of the intersection of two or more geometric shapes in a plane (7-4.2). They also illustrate the cross section of a solid (7-4.3). In eighth grade, students use ordered pairs, equations, intercepts and intersections to locate points and lines in a coordinate plane (8-4.2). Taxonomy Level Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts

• Intersection • Geometric shapes


for students to:

• Understand the meaning of intersection • Understand the result of a line crossing another line is a point • Understand the result of a line crossing a circle is two point • Explain the results from the intersection of any 2 figures that they have

studied. • Understand the orientation of the shapes effect the number of intersections.


5


for students to:

None noted Student Misconceptions/Errors It may be difficult for students to visualize the orientation of shapes. Instructional Resources and Strategies Using models and pictures will aid student as they try to visualize these intersections. For example, a pen and piece of paper may be used to demonstrate what happens when a plane and a line intersect. Assessment Guidelines The objective of this indicator is to explain which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To explain is to construct a cause and a sentence structured like “The point of intersection of _____ is ______ because ____.” The learning progression to explain requires students to recall the characteristic of points, lines and planes. They explore concrete and pictorial models of the intersection of geometric shapes and analyze them to determine where and how the shapes intersect. Students use deductive reasoning (7-1.3) to generalize mathematical statements (7-1.5) related to the intersection of certain shapes (line and circle, line and plane, circle and circle, etc…). They use

correct and clearly written or spoken words to communicate their understanding (7-1.6).


6

Indicator 7-4.3

Illustrate the cross-section of a solid.


In fifth grade, students translated between two-dimensional representations and three-dimensional objects (5-4.4).

In seventh grade, students illustrate the cross section of a solid (7-4.3).

Taxonomy Level


Vocabulary

• Three-dimensional • Two-dimensional • Isometric • View Point • Illustrate • Vertical plane • Horizontal plane • Solid


for students to:

• Understand and visualize horizontal and vertical cross planes • Determine the cross sections for cubes, prisms, sphere and cylinders, cones

and pyramids • Understand the connection between the three dimensional object and its two

dimensional cross section (sphere – circle, rectangular prism – rectangle, etc..)



7

• Explore cross sections using concrete and pictorial models For this indicator, it is not essential

for students to:

• Focus on cross sections that are not vertical or horizontal. No cross sections cut at a slant.


• Students often cannot visualize three- dimensional shapes from two-dimensional drawings.


• It may be helpful for students to think of cross sections as “deconstructing” the layers of a three-dimensional object. For example, to illustrate the cross section of a rectangle, students may build a rectangle using interlocking cubes. Then the students might illustrate on isometric dot paper the original rectangle and a view of one of the “layers” or cross sections that make up the rectangle. The ability to do this will enable students to develop, justify, and understand formulas (such as area, surface area, volume, etc.) that are used in regards to two- and three-dimensional figures.

• Another strategy involves using real world objects. For example, an orange can be used for a sphere, a small rectangular cake can be used for the rectangular prism and a stack of cookies can be used to illustrate a cylinder.

• When engaging in discussion, it is important for students to understand that the concrete models and pictorial illustrations are the result of the intersection of a plane in a section of the geometric shape. (This stays true to the definition of a cross-section- the intersection of a plane and a geometric solid.)

• Teaching Student-Centered Mathematics Volume 3 Grades 5-8

, Van de Walle, pp. 224, 225.

• Getting Solid with Shapes http://www.indianastandards.org/files/math/math_8_4_3.pdf

Assessment Guidelines The objective of this indicator is to illustrate, which is in the “understand procedural” knowledge cell of the Revised Taxonomy. To illustrate is to find specific example or illustrate of a concept; therefore, students explore a variety of example of cross section in order to understand how cross sections are found. The learning progression to illustrate requires students to recall the characteristics of cubes,

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8

rectangular prisms, cylinders and spheres. They analyze a variety of solids that are orientated vertically and model them with concrete objects in order to develop a mental picture of these three dimensional shapes. Students explore the result of cutting horizontal and vertical sections and generalize mathematical statements (7-1.5) about the relationship between the solid and its cross section. The students pose follow-up questions to the teachers and their classmates in order to prove or disprove their conjectures (7-1.2). They generalize connection among a variety of representational forms (concrete, pictorial and real world) to deepen their conceptual understand of this relationship. Students use

correct and clearly written and spoken words to communicate their understanding (7-1.6).


9

Indicator 7-4.4

Translate between two and three dimensional representations of compound figures.


In fifth grade, students explored methods for translating between two-dimensional representations and three-dimensional objects (5-4.4).

In seventh grade, students translate between two and three dimensional representations of compound figures (7-4.4).

Taxonomy Level


• Compound Figure Vocabulary

• Perspective • Translate • Cube • Sphere • Cylinder • Prism • Pyramid • Cone • Rectangle • Square • Triangle • Circle


for students to:

• Recall the characteristics of two and three dimensional shapes (see Key Concepts)



10

• Identify from concrete and pictorial models the view points that give the figure its overall shape.

• Identify from concrete and pictorial models the overall shape when provided with the perspective (front, top and side views)


for students to:

• Illustrate the nets for a given three-dimensional shape. • Construct a three-dimensional shape when given its net.


• Students may think that compound figures are constructed of different shapes. Clarify that compound figures may be composed by repeating any given shape (stacks of cubes). For example,


• Extensive modeling with concrete objects needs to be done in order for the students to develop a mental picture of compound three-dimensional shapes and the two-dimensional viewpoints that give the figure it’s overall shape and vice versa.

• Using Cubes and Isometric drawings http://illuminations.nctm.org/LessonDetail.aspx?id=U166 • Teaching Student-Centered Mathematics Volume 3 Grades 5-8

, Van de Walle, pp. 224, 225.

• Guess The View http://www.fi.uu.nl/toepassingen/00198/toepassing_wisweb.en.html

Assessment Guidelines The objective of this indicator is to translate, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To translate is to change from one form to another; therefore, students develop a conceptual understanding of the relationship between two dimensional compound shapes and their three dimensional representation. The learning progression to translate requires

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11

students to recall the characteristics of two and three-dimensional shapes and their two-dimensional nets. Students use concrete and pictorial models to explore how two-dimensional shapes work together and predict their perspectives using inductive and deductive reasoning (7-1.3). Students explore front, top and side views. Students generalize connections (7-1.7) between two dimensional and three dimensional representations using correct and clearly written words to communicate (7-1.6) their understanding. They use

that understanding to translate between two and three-dimensional representations of compound figures.


12

Indicator 7-4.5

Analyze the congruent and supplementary relationships –specifically, alternate interior, alternate exterior, corresponding, and adjacent- of the angles formed by parallel lines and a transversal.


In third grade, students classified lines and line segments as either parallel, perpendicular or intersecting lines (3-4.3).

In the seventh grade, students analyze the congruent and supplementary relationships –specifically, alternate interior, alternate exterior, corresponding, and adjacent- of the angles formed by parallel lines and a transversal (7-4.5).

Taxonomy Level

Cognitive Dimension: Analyze Knowledge Dimension: Conceptual Key Concepts

Vocabulary

• Parallel • Transversal • Alternate • Corresponding • Intersecting • Congruent • Supplementary • Perpendicular • Adjacent


for students to:

• Understand the term congruent angles.



13

• Understand the term supplementary angles. • Recall what two parallel lines cut by a transversal looks like • Understand and identify the terms alternate exterior, alternate interior,

corresponding, and adjacent in regards to the angles formed by parallel lines and a transversal.

• Find the value of a missing angle For this indicator, it is not essential

for students to:

• Use multiple sets of parallel lines and transversals in one drawing. Student Misconceptions/Errors Students will think that angles are congruent when there are no parallel lines present. Emphasize that angles (alternate exterior, alternate interior, corresponding, and adjacent angles) can only be congruent when they are formed by parallel lines being intersected by a transversal. Students may be confused when the transversal is slanted differently. Exploring examples where the transversal is increasing from left to right and decreasing from left to right will help with this confusion.


Students need to formulate their own conclusions in regards to the angles formed by two parallel lines and a transversal through guided investigation before the formal definitions are introduced. Label the angles 1,2, 3, 4, etc… They measure the angles formed by these lines, and from their findings, make conjectures and draw conclusions about which angles are congruent( a concept taught in fifth grade) and which angles are supplementary( a concept taught in sixth grade). Once the students have a good understanding of which angles are congruent and/or supplementary, then the formal definitions can be introduced.

Assessment Guidelines The objective of this indicator is to analyze, which is in the “analyze conceptual” knowledge cell of the Revised Taxonomy. To analyze means to break down materials into its constituent parts and determine how the parts relate to one another and the overall structure; therefore, students determine the relationship between and among angles in the parallel line/transversal diagram. The learning progression to analyze requires students to recall the diagram illustrating two parallel lines cut by a transversal. Given a diagram, students explore the relationship by measuring angles. They generalize connections (7-1.7) between angles using correct and clearly written or spoken words (7-1.6). They explore a


14

variety of examples to evaluate their conjectures (7-1.2) using inductive and deductive reasoning (7-1.3). They use these connections to generalize mathematical statements (7-1.5) that summarize the relationship between angles. At this point, students connect

these relationships to the terms alternate interior, alternate exterior, corresponding and adjacent angles.


15


Indicator 7-4.6

Compare the areas of similar shapes and the areas of congruent shapes.

Continuum of Knowledge:

In sixth grade, students compared the angles, side lengths, and perimeters of similar shapes (6-4.7)

In seventh grade, students compare the area of similar and congruent shapes (7-4.6), explain the proportional relationships among similar shapes (7-4.7) and apply procedures to find missing attributes of similar shapes (7-4.8).

In eighth grade, students apply dilations to squares and other polygons to form similar shapes on the coordinate system (8-4.4).

Taxonomy Level

Cognitive Dimension: Understand Knowledge Dimension: Conceptual

Key Concepts

• Corresponding sides

Vocabulary

• Corresponding angles • Similar shape • Congruent shape • Proportional reasoning

Instructional Guidelines

For this indicator, it is essential

• Identify the corresponding angles and sides of similar and congruent figures.

for students to:

• Know the difference between a similar and congruent figure. • Know that the area of congruent shapes is equal while the area of similar

shapes is proportional but not equal.


16

• Use proportional reasoning to describe relationships between the areas of similar shapes

• Understand how increasing the side lengths affect the area- Example if you triple the side length the area is 9 times the original area.


• To find the length of a missing side using a proportion.

for students to:

• To find the measure of a missing angle. Student Misconceptions/Errors

Students often think the increase of the side length will be the same for the increase in the area. They do not make the connection that if the side length is multiplied by 3 or tripled, then the area will be increase by the square of that number or in this case will be nine times the original area.


• Students will compare the areas of similar shapes and the areas of congruent shapes. Students should be given the opportunity to discover that the areas of congruent shapes are equal whereas the areas of similar shapes are not.

• Although not required, this indicator may be extended to allow students to discover and understand that the area of the larger of two similar shapes will be the area of the smaller shape multiplied by the square of the scale factor needed to create the larger similar shape. This may be difficult for students to initially conclude; therefore, numerous examples should be done to help students see this relationship.

• The use of isometric dot paper is a great strategy to show the students the relationships between similar shapes and area.

Assessment Guidelines

The objective of this indicator is compare, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To understand is to construct meaning; therefore, students develop a conceptual understanding of the area of similar and congruent shapes using a variety of examples. The learning progression to compare requires students to recall the relationships among attributes of similar and congruent shapes. Using a variety of concrete and pictorial examples, students explore relationships among the area of similar and congruent shapes. They also explore how the area is affected by changes in the dimensions. Students generalize connections (7-1.7) and generalize mathematical statements (7-1.5) based on their observations. They use these generalizations to compare areas of a variety of shapes.


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Indicator 7-4.7

Explain the proportional relationship among attributes of similar shapes.


In sixth grade, students compared the angles, side lengths, and perimeters of similar shapes (6-4.7) and classified shapes as similar (6-4.8).

In seventh grade, students compare the area of similar and congruent shapes (7-4.6). They explain the proportional relationships among similar shapes (7-4.7) and apply procedures to find missing attributes of similar shapes (7-4.8).

In eighth grade, students apply dilations to squares and other polygons to form similar shapes on the coordinate system (8-4.4). These standards will also be addressed again in high school geometry.

Taxonomy Level

Cognitive Dimension: Understanding Knowledge Dimension: Conceptual

Key Concepts


Vocabulary

• Corresponding angles • Proportional relationship • Proportion • Cross products



• Identify the proportional relationships of polygons.

for students to:

• Set up proportions using corresponding sides. • Understand that cross products of a proportion should be equal to prove that

two polygons are similar.


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• Understand that sides that correspond are connected by corresponding angles that are congruent.

• Understand that corresponding angle measures are congruent while corresponding sides are proportional


• To find the side length of a missing side

for students to:

• To find the measure of a missing angle Student Misconceptions/Errors

Students may forget that they should have a correspondence between sides and angles.


None noted


The objective of this indicator is to explain which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To explain means to construct a cause and effect model; therefore, students should show their conceptual understanding of these proportional relationships is that model. The learning progression to explain requires students to recall and understand the characteristics of similar shapes. Students analyze shapes and create a correspondence between angles and sides. They use these correspondences to set up possible proportional relationships. Students use their understanding of the cross product to evaluate their conjectures then explain and justify

their answers using correct and clearly written or spoken words (7-1.6).


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Indicator 7-4.8

Apply proportional reasoning to find missing attributes of similar shapes.


In sixth grade, students compared the angles, side lengths, and perimeters of similar shapes (6-4.7) and classified shapes as similar (6-4.8).

In seventh grade, students compare the area of similar and congruent shapes (7-4.6). They explain the proportional relationships among similar shapes (7-4.7) and apply procedures to find missing attributes of similar shapes (7-4.8).

In eighth grade, students apply dilations to squares and other polygons to form similar shapes on the coordinate system (8-4.4). These standards will also be addressed again in high school geometry.

Taxonomy Level

Cognitive Dimension: Apply Knowledge Dimension: Procedural

Key Concepts


Vocabulary

• Corresponding angles • Proportional relationship • Proportion • Cross products



• Understand the characteristics of similar shapes

for students to:

• Determine corresponding relationships between angles and sides


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• Solve a proportion • Understand proportional reasoning • Use proportions to find the measure of a missing side length. • Use proportions to find the measure of a missing angle


• To find the measure of a missing angle based on the sum of the angles in the polygon.

for students to:


Students often think that all shapes have the same orientation. Students do not focus on the concept that corresponding sides are connected by the corresponding angles.


• Once again, it is extremely important that students have made the connections in regards to similar figures that corresponding angle measures are congruent while corresponding sides are proportional.

• Seventh grade students must have an in-depth understanding of when and how to apply proportional reasoning to solve various types of problems. Although students set up a proportion to find missing attributes, they should also communicate how they would solve the problem using proportional reasoning. A sample explanation could be “since this side is twice as big as its corresponding side in order to keep the proportional relationship the other side must be twice as much too so it equals 8.”


The objective of this indicator is to apply, which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Although the focus of the indicator is to apply, the learning progression should integrate strategies to support conceptual as well as procedural knowledge of proportional reasoning. The learning progression to apply requires students to recall the characteristics of similar shapes. They also recall and understand how determine corresponding parts of similar shapes. Students use their understanding of the relationships between corresponding parts to describe proportional relationships. They describe these relationships verbally using proportional reasoning and symbolically using a proportion with an unknown. Students evaluate their answers (7-1.2) and use correct and clearly written or spoken words (7-1.6) to justify their answers using their understanding of proportional reasoning.


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Indicator 7-4.9

Create tessellations with transformations.


In sixth grade, students identified the transformations used to move a polygon from one location to another on the coordinate plane (6-4.5). Students also explained how the transformation would affect the original location of the polygon in the coordinate plane (6-4.6).

In the seventh grade, students use transformations to create tessellations (7-4.9).

Taxonomy Level

Cognitive Dimension: Create Knowledge Dimension: Conceptual

Key Concepts

• Transformation

Vocabulary

• Rotation • Translation • Reflections • Tessellation



• Use transformations (translations, reflections, and rotations) to change a polygon to a new shape by cutting sections out of one or two of the sides and using a transformation to place this section on the other side.

for students to:

• Use the transformed polygon as a stencil to create their own tessellation and explain how they transformed the polygon (i.e. identify the transformation used to create the stencil).

• Identify the transformation used to create a given tessellation. • Create tessellations using combinations of regular polygons and rectangles.


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• Create a tessellation using non-regular polygons (other than rectangles)

for students to:


• None noted


• This is the students’ first introduction to tessellations. Tessellations (the covering of a plane without overlaps or gaps) are formed by transformations such as translations, reflections, and rotations. The emphasis should be on the transformations used to create the tessellation and the relationship among the polygons that can be used to tessellate (or tile) a plane (piece of paper).

• A great investigational lesson would use a square (sticky note size or smaller) or any regular polygon and let the students transform the polygon using a translation, reflection, or rotation. Students will change one or two sides of the square and then use this transformed square as a stencil to create a tessellation. The student needs to explain and illustrate how the polygon was transformed and explain how this affected the final tessellation. A guided lesson for this would be essential, tell the students to create a tessellation using each type instead of having them try to create all in one day. Students can also add details inside their transformed image to make their tessellation appear more Escher like.

• Students should also be able to identify what transformation was used to create a tessellation from a given example. Escher examples may be used but can get complicated and may be hard to identify.


The objective of this indicator is to create, which is in the “conceptual knowledge” of the Revised Taxonomy. To create is to put together elements to form a new, coherent whole or to make an original product; therefore, students use their understanding of transformations to create tessellations. The learning progression to create requires students to recall the definitions of translation, reflection, and rotations and to identify when they are being used. Students explore various strategies that allow them to develop an understanding of how to create a tessellation using the various transformations. Students apply their conceptual knowledge of transformations to create tessellations. Students also identify the transformations used to create a given tessellation. Students should use these


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procedures in context by examining real world examples (7-1.7) such as tiles or patterns as opposed to only creating tessellations from polygons. They use

correct and clearly written or spoken words to communicate their understanding of the transformations used to create the tessellation (7-1.6).


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Indicator 7-4.10

Explain the relationship of the angle measurements among shapes that tessellate.


In sixth grade, students identified the transformations used to move a polygon from one location to another on the coordinate plane (6-4.5). Students also explained how the transformation would affect the original location of the polygon in the coordinate plane (6-4.6).

In the seventh grade, students use transformations to create tessellations (7-4.9).

Taxonomy Level

Cognitive Dimension: Understand Knowledge Dimension: Conceptual

Key Concepts

• Vertex

Vocabulary

• Regular polygons • Sum of the angle • Interior angle • Plane • Tessellation

•

Mathematical Formula

nn 180)2( −



• Find the measure of an interior angle of a polygon

for students to:


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• Identify if a regular polygon will form a tessellation by itself or if it can be used in a combination to create a tessellation.

• Understand that in order for regular polygons to tessellate, the sum of the measure of the angles surrounding appoint (at a vertex) must 360 degrees.

• Use the formula n

n 180)2( − to find the measure of one interior angle.

(n-2)180 will tell the students the sum off all the angles. N= number of sides.

• Know that tessellations cover a plane without overlaps or gaps using congruent figures or a combination of congruent figures


• Identify if a non-regular polygon will form a tessellation.

for students to:


• None noted


• One strategy to teach finding the sum of angles and the measure of one angle of a regular polygon is by using the chart A below. Another strategy is to use Chart B. Both of these strategies will require guidance from the teacher. The objective is for students to discover that in order for regular (all sides are the same length) polygons to tessellate, the sum of the measures of the angles surrounding a point (at a vertex) must be 360°. This sum can be attained by using a combination of different regular polygons or just several of the same polygon. Using regular polygon manipulatives and the following chart may be helpful.


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Chart A: Regular Polygon - # 0f sides

# of Triangles created from 1 vertex

Formula:___________

Sum of degrees for all angles of the polygon

Formula:__________

# in degrees in 1interior angle of the polygon.

Formula:_________

Will this polygon form a pure tessellation?

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon/Septagon

Octagon

Nonagon

Decagon

Dodecagon

20-gon

30-gon

Chart B: Shape Triangle Square Hexagon Octagon

Measure of one

interior angle

60°

90°

120°

135°

Combinations that total 360° and therefore will tessellate.

Number 6 60 (6) = 360

of 4 1 60 (4) + 120 = 360

each 1 2 90 (1)+ 135 (2) = 360

shape 3 120 (3) = 360

Used 1 2 1 60(1)+90(2) +120(1) = 360


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The objective of this indicator is to explain, which is in the “understand conceptual” of the Revised Taxonomy. To explain is to construct a cause and effect model; therefore, students summarize relationships among angles that tessellate using that model. The learning progression to explain requires students to recall the meaning of tessellate and how to create a tessellation. Students also recall the characteristics of regular polygons and how to find the measure of one interior angle of a polygon. They use this knowledge to explore relationships among the angles of a variety of regular polygons. They generalize connections (7-1.7) and generalize mathematical statements (7-1.5) explaining those connections. They use correct and clearly written or spoken words to communicate their understanding of these relationships to their classmates and teacher (7-1.6).

Measurement Seventh Grade

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Standard 7-5: The student will demonstrate through the mathematical processes an understanding of how to use ratio and proportion to solve problems involving scale factors and rates and how to use one-step unit analysis to convert between and within the U.S. Customary System and the metric system. Indicator 7-5.1 Use ratio and proportion to solve problems involving scale factors and rates. Continuum of Knowledge In sixth grade students used proportions to determine unit rates (6-5.6). Sixth grade students also used a scale to determine distance. In seventh grade, students extend their understanding of and use of proportional reasoning (unit rates and scale) to solve problems using ratio and proportion involving scale factors and rates (7-5.1). In eighth grade, students use proportional reasoning and the properties of similar shapes to determine the length of a missing side (8-5.1). Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts

• Ratio Vocabulary

• Rate • Proportion • Scale • Unit rate


for students to:

• Understand the meaning of ration • Understanding the meaning of proportion • Set up a ratio • Set up a proportion

• Interpret their answers. For example, hourmiles

5250 means he drove 250

miles in 5 hours.


2

• Work with answers that are in whole number, decimal or fractional form

• Use an appropriate strategy to solve the proportion For this indicator, it is not essential

for students to:

• Solve problems that involve missing pieces of similar figures (8-5.1). Student Misconceptions/Errors Instructional Resources and Strategies

• This indicator should not only focus on setting up proportions and using the algorithm to solve for the missing piece, but also, focus on building students’ ability to use proportional reasoning. Proportional reasoning is hard to teach and should be developed over time using a variety of activities.

• Explore examples where scale and rates are used in the real-world (i.e. blueprints, speed, models)

• A hands-on activity is a scale drawing project. This can be done outside of school or as an in-class project. Give students a variety of coloring books (or the student can bring in a picture). Once they have selected their pictures, you will need to superimpose a grid on their picture. They will need your help will this next part because it requires measuring and partitioning off their squares. Given the dimensions of the paper (posterboard, 11 x 17 paper, etc..) on which they will be drawing, students determine what size squares they will need so their enlarged picture will fit on the paper. They determine their scale factor. Next, they have to numbers the squares on their picture and number the squares on their drawing paper. They have to redraw each square so that it is proportional to what is in the square on their picture. The key is that they only drawn one square at a time.

• The introduction and use of the procedural concept of the cross-product algorithm should only be introduced after students have had considerable amount of time with developing and experimenting with more intuitive and conceptual methods for solving proportions.

• One Inch Tall by Shel Silverstein (poem) (Here students can use ratios

and proportionality to make decisions about the statements made in the poem based on mathematics.)

Assessment Guidelines The objective of this indicator is use, which is in the “apply procedural” cell of the Revised Taxonomy. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem or problem situation. Although the focus is to gain computational fluency with problems involving


3

the use of proportions to solve problems with scale and rates, the learning progression should integrate strategies to enhance both conceptual and procedural knowledge. The learning progression to use requires students recall the definition of ratio and proportion and understand how to use proportions (equivalent ratios) to solve simple problems involving unit rates. Students explore a variety of situations that involve scale factors and rates and generalize connections among real-world situations (7-1.7). They develop strategies for solving problems and explain and justify their strategies using their understanding of proportional reasoning. They use these strategies (conceptual and procedural) to generate and solve other problems. They explain and justify their answers to their classmates and teacher using correct and clearly written or spoken (7-1.6).


4

Standard 7-5: The student will demonstrate through the mathematical processes an understanding of how to use ratio and proportion to solve problems involving scale factors and rates and how to use one-step unit analysis to convert between and within the U.S. Customary System and the metric system. Indicator 7-5.2 Apply strategies and formulas to determine the surface area and volume of the three dimensional shapes prism, pyramid, and cylinder. Continuum of Knowledge In sixth grade, students generated strategies to determine the surface area of a rectangular prism and a cylinder (6-5.3). In seventh grade, students apply strategies and formulas to determine the surface area and volume of the three dimensional shapes prism, pyramid, and cylinder (7-5.2). In eighth grade, students apply strategies and formulas to determine the volume of the three-dimensional shapes cone and sphere (8-5.3). Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts

• Surface Area Vocabulary:

• Volume • Three-Dimensional • Base • Height • Lateral Area • Cylinder, Prism, and Pyramid (review characteristics) • Area of Base


for student to:

• Understand the concept of surface area as the areas of all the faces • Understand the concept of height of a solid • Be able to fluently calculate with fractions, decimals, and exponents.


5

• Know the estimated equivalent of pi as 3.14 and722 .

• Be able to calculate circumference • Be able to find the area of squares, rectangles, triangles, trapezoids. • Understand the concept of volume—cubic units


for students to:

• Calculate using non-equilateral triangles, pentagons, hexagons, etc. Student Misconceptions/Errors Students often confuse the concept of surface area and volume. Teachers need to build conceptual knowledge for these two concepts through using hands-on models and visual explorations. For example, nets with grids on the faces could be used to build the definition of surface area, and unifix cubes could be used to conceptualize the definition of volume. The indicator does not specify the types of prisms, pyramids and cylinders; therefore, they experiences should include a variety of examples i.e. rectangular prism, triangular prism, triangular pyramids, etc.. Instructional Resources and Strategies

• Review the characteristics of each solid and allow students time to investigate the role the height plays in the calculation of volume and surface area.

• Use models, nets, cubes and other hands-on explorations to investigate the concept of surface area and volume.

• Explore various investigations using manipulatives (unifix cubes, blocks, etc) to discover the concept of the formulas for surface area and volume.

• Use 3-D models that have capabilities to pour water/rice/beans/etc. into them to show the relationships among volume formulas.

• Counting on Frank by Rod Clement is a great picture book that offers scenarios for students to explore and apply their knowledge and understanding of volume. Teachers could extend the book and connect it to surface area with their students.

• Students should have opportunity to set-up and solve multiple problems and problem situations. They also need to have opportunities to use a calculator to explore more challenging problems.



6

The objective of this indicator is to apply, which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Procedural knowledge is knowledge of specific steps or strategies to solve a problem or problem situation. Although the focus is to gain computational fluency with calculating surface area and volume of prisms, pyramids, and cylinders, the learning progression should integrate strategies to enhance both conceptual and procedural knowledge. The learning progression for finding surface area requires students to recall the characteristics of the faces of rectangular and triangular prisms, rectangular and triangular pyramids, and cylinders. They also need to understand area of rectangles, circles, squares, and triangles, as well as, circumference of circles. Students use their conceptual knowledge of area to translate their understanding of concrete representations to the symbolic form and be able to apply their understanding to a 3-D picture of each type of solid (rectangular prism, square pyramid, triangular pyramid with equilateral base, and cylinders). Students need to be able to generalize and communicate the formula for surface area based on investigations/hands-on activities to see relationships for the symbols in the formulas. Students should also use correct and clearly written or spoken words to explain their reasoning for their answers (7-1.6). Students should apply these procedures in context as opposed to only rote computational exercises and generalize

The learning progression for volume of 3-D figures requires students to

connections among representational forms and real world situations (7-1.7).

recall the characteristics of the faces of rectangular and triangular prisms, square and triangular pyramids, and cylinders. They understand that the concept of volume is the number of cubic units to fill a space. Students generalize and communicate the formula for volume based on investigations/hands-on activities to see relationships among the solids and connections to the symbols in the formulas (7-1.4). Students should also use correct and clearly written or spoken words to explain their reasoning for their answers (7-1.6). Students should apply these procedures in context as opposed to only rote computational exercises and generalize

connections among representational forms and real world situations (7-1.7).

Standard 7-5: The student will demonstrate through the mathematical processes an understanding of how to use ratio and proportion to solve problems involving scale factors and rates and how to use one-step unit


7

analysis to convert between and within the U.S. Customary System and the metric system. Indicator 7-5.3 Generate strategies to determine the perimeters and areas of trapezoids. Continuum of Knowledge In sixth grade, students applied strategies and procedures to estimate the perimeters and areas of irregular shapes (6-5.4). Also, students in sixth grade applied strategies and procedures of combining and subdividing to find perimeters and areas of irregular shapes (6-5.5). In seventh grade, students generate strategies to determine the perimeters and areas of trapezoids (7-5.3). This is the first time students will be introduced to area of trapezoids. In eighth grade, students apply formulas to determine the perimeters and areas of trapezoids (8-5.5). Taxonomy Level Cognitive Dimension: Create Knowledge Dimension: Conceptual Key Concepts

• Perimeter Vocabulary:

• Trapezoid • Area • Base • Height


for students to:

• Understand the concept of perimeter • Understand the concept of area • Recall the characteristics of a trapezoid • Recognize the various forms of trapezoids and rotations of the figure • Recall and understand how to calculate areas of other shapes (i.e.

rectangles, squares, parallelograms and triangles) For this indicator, it is not essential for students to:


8

• Gain computational fluency in these procedures

Student Misconceptions/Errors A common error made by students when using formulas comes from no conceptual understanding of the meaning of height in geometric figures. Before using formulas involving height, students should discuss the meaning of height of a geometric figure and be able to identify where a height could be measured. Instructional Resources and Strategies

• After giving students a picture of a trapezoid with various lengths, students can work with partners to explore possible strategies to calculate the area of a trapezoid. The teacher will serve as a facilitator, asking questions to get students to find the area or perimeter. Then after some time of exploration, and students have their strategies, they will need to test their strategy on various trapezoids to see if works. Students will share with the class how they determined the perimeter and area of the trapezoid. Students may recall that from their sixth grade experience of sub-dividing an irregular shape into several known shapes. This may be a strategy that they could try.

• Following estimating perimeters and areas of trapezoids, give students a lab investigation to discover the formula for finding area of trapezoids. Have students cut out two identical trapezoids, put them together to form a parallelogram, and relate the area of the parallelogram formed to the area of the trapezoid. Have students generate problems for other students to solve, and then the students will exchange problems for partners to solve.

• Teaching Student-Centered Mathematics Grades 5-8

Teachers can provide students with various manipulatives: pattern blocks, scissors, colored pencils etc. to be available during their exploration.

by John A. Van de Walle

Assessment Guidelines The objective of this indicator is to generate strategies, which is in the “create conceptual” knowledge cell of the Revised Taxonomy table. To create means to reorganize elements (area of perimeter of other polygons) into a new structure (trapezoids). The learning progression to generate requires students to recall formulas for area of squares, rectangles, parallelograms, and triangles. They recall the definition of height of a geometric shape and understand how to locate the height. Students work cooperatively to generate strategies for determining the area of a trapezoid. Students share


9

their strategies with the class and generalize mathematical statements (7-1.5) about their strategy using clear and correct spoken words to communicate their understanding (7-1.6). Students evaluate their strategies using a variety of trapezoids to prove or disprove their statement (7-1.2). After discussing students’ strategies, students may revise their mathematical statement based on their observations. Students apply their strategies and a formula to determine the area and perimeters of trapezoids through various problem situations and generate

problems for peers to solve (7-1.1).


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Standard7-5: The student will demonstrate through the mathematical processes an understanding of how to use ratio and proportion to solve problems involving scale factors and rates and how to use one-step unit analysis to convert between and within the U.S. Customary System and the metric system. Indicator 7-5.4 Recall equivalencies associated with length, mass and weight, and liquid volume: 1 square yard = 9 square feet, 1 cubic meter = 1 million cubic

centimeters, 1 kilometer = 85 mile, 1 inch = 2.54 centimeters; 1 kilogram =

2.2 pounds; and 1.06 quarts = 1 liter. Continuum of Knowledge In fourth grade, students recalled basic conversion facts within the customary system for length, weight and capacity (4-5.8). They used equivalencies to convert units of length, weight, and capacity within the Customary System (4-5.3). In fifth grade, students focus on the metric system recalling facts (5-5.8) and converting within the metric system (5-5.3). In seventh grade, students will recall equivalencies associated with length, mass and weight, and liquid volume (7-5.4). Also students use one step unit analysis to convert within and between the U.S. Customary System and metric system (7-5.5). In eighth grade, students use multi-step unit analysis to convert within and between the Customary and metric systems (8-5.7). Taxonomy Level Cognitive Dimension: Remember Knowledge Dimension: Factual Key Concepts

• Units and Equivalencies from the indicator • Equivalent expressions

Instructional Guidelines For this indicator, it is essentia

l for students to be able to:

• Recall measurement facts listed in the indicator • Explore real world model of each to support retention


11

• Generate common references for these equivalencies. For example, For this indicator, it is not essential

for students to:

• Do any problems for this indicator Student Misconceptions/Errors Students have difficulty knowing when to use appropriate units of measure such as linear units, square units or cubic units. Instructional Resources and Strategies

• Allow students to participate in hands-on activities that visually demonstrate what each measurement looks like—For example, one section on their index finger is about 1 inch. Also have student feel the weight of something that weighs 2.2 pounds, etc..

• Memory cards and games can be used to support retention.

Assessment Guidelines The objective of this indicator is recall, which is in the “factual knowledge” cell of the Revised Taxonomy. Although the focus of the indicator is to recall, integrating hands-on activities and real world examples help to support retention. The learning progression to recall requires students to understand that each fact is a symbolic form (in and cm or pounds and kilogram) that represents the same relationship or quantity (7-1.4). Students explore hands-on or real world example, where appropriate, to generalize the connection among a variety of representational forms (7-1.7) of these equivalencies. They should develop

personal and meaning strategies for recalling these facts.


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Standard 7-5: The student will demonstrate through the mathematical processes an understanding of how to use ratio and proportion to solve problems involving scale factors and rates and how to use one-step unit analysis to convert between and within the U.S. Customary System and the metric system. Indicator 7-5.5 Use one-step unit analysis to convert between and within the U.S. Customary System and the metric system. Continuum of Knowledge In fourth grade, students recalled basic conversion facts within the customary system for length, weight and capacity (4-5.8). They used equivalencies to convert units of length, weight, and capacity within the Customary System (4-5.3). In fifth grade, students focus on the metric system recalling facts (5-5.8) and converting within the metric system (5-5.3). In seventh grade, students will recall equivalencies associated with length, mass and weight, and liquid volume (7-5.4). Also students use one step unit analysis to convert within and between the U.S. Customary System and metric system (7-5.5). In eighth grade, students use multi-step unit analysis to convert within and between the Customary and metric systems (8-5.7). Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts

• Unit analysis Vocabulary

• Equivalencies • Unit cancellation (cancelling units)

Abbreviations for length, weight/mass, and liquid volume for U.S. customary and metric system units

Mathematical Notation/Symbols:


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for students to:

• Understand proportional reasoning • Multiplying fractions • Simplify expression • Understand that each equivalency when written as a fraction is a form

of one • Use equivalencies to convert between systems • Set up ratios to convert using unit analysis:

For example, if there are 2.54 cm to 1 inch, how many centimeters are in a foot? By writing each in fraction format (horizontally), students can see that all units cancel except the final ones in the answer.

2.54 cm x 12 in. = 30.48 cm 1 in. 1 ft. 1 ft.

= 30.48 cm in a foot

Another example: 10 yd to feet

10 yd x 3 ft = 10 x 3 ft 1 1 yd 1

= 30 ft


for students to:

• Memorize the equivalencies • Do more than a one-step unit-analysis problems (8th

grade)

Student Misconceptions/Errors Students may have trouble with placing their units in the correct position of the fraction to get their units to “cancel”. Instructional Resources and Strategies In the non-essentials, it states that students do not need to memorize the equivalencies in order to meet this indicator. Although it would be beneficial for students to recall the equivalencies, it is not essential because the focus of the indicator is on the process of unit analysis

not on memorizing facts. The work of students should be on gaining computational fluency with the process of setting up and simplifying the problem. Students will become frustrated because they can’t recall the equivalencies and will be distracted from the central focus.

One strategy that may be used to address this issue is to give students an assessment on recalling their facts (focus on retention) and a separate assessment where students are given the equivalencies and asked to do conversions (focus on conversions).


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In science, unit analysis is typically called dimensional analysis; therefore, students should be familiar with both terms. Assessment Guidelines The objective of this indicator is use, which is the “apply procedural” knowledge cell of the Revised Taxonomy. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem. The focus is to gain computational fluency with conversions within and between the US Customary and metric system. The learning progression to use requires students to understand the relationship between the equivalencies and a form of one. Students explore teacher generated examples to investigate what happens to the ratios when they are multiplied (what cancels or doesn’t cancel). They generalize mathematical statements (7-1.5) summarizing a strategy that can be used to set up and simplify a problem. Students use their understanding of proportional reasoning to explain and justify their answers using correct and clearly written or spoken words. They also understand

the original measurement and the converted measurement are different symbolic forms of the same quantity (7-1.4).

Data Analysis and Probability Seventh Grade

1

Standard 7-6: The student will demonstrate through the mathematical processes an understanding of the relationships between two populations or samples.

Indicator 7-6.1

Predict the characteristics of two populations based on the analysis of sample data.


In sixth grade, students predicted the characteristics of one population based on the analysis of sample data (6-6.1).

In seventh grade, students predict the characteristics of two populations based on the analysis of sample data (7-6.1).

In eighth grade, students will generalize the relationship between two sets of data by using scatterplots and lines of best fit (8-6.1).

Taxonomy Level

Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts Vocabulary

• Population • Data • Analyze • Sample data • Test data • Random sample • Control sample • Shape of the data


• Make predictions from data in varying formats. • Translate data to a graph or a picture • Understand that the prediction from the sample data is an estimation for the

population • Make predictions based on the shape of the data (central tendency, spread

of the data and outliers)


2

• Compare the distribution of the data (shape) • Observe trends in the data • Understand that the differences in spread of the two data sets are important

in comparing the sets • Justify their predictions using mathematical reasoning

For this indicator, it is not essential for students to:

• Analyze data in scatter plots. • Compare data sets by using summary statistics. • Compare data sets that are in different representational forms


• Students may assume that they are making predictions when they are actually guessing.

• Students do not make their predictions based on mathematical reasoning.


• Having students analyze real world data that is relevant to their lives, it is an excellent strategy to build conceptual understanding and engage students. Students could plan and design experiments to collect data. An example could be data on sneaker prices from two stores. Students analyze the data and make inferences and predictions related to the expected cost of a pair of sneaker.

• NCTM: http://www.nctm.org/profdev/content.aspx?id=11688 – Middle School Data Resources complied by the National Council of Teachers of Mathematics.

Assessment Guidelines The objective of this indicator is to predict, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To predict means to draw logical conclusion from presented information. The learning progression to predict requires students analyze two sets of data and generate conjectures about the population. They understand that it is sometimes difficult to do so from just analyzing the numbers. They translate their data to another form (graph or picture). They understand that each is a distinct symbolic form that represents the same relationship (7-1.4) and generalize connections (7-1.7) among these representational forms. They make observations about the shapes and proximity of the data in order to make reasonable comparisons and predictions based on those observations. They use inductive and deductive reasoning (7-1.3). Students explore a variety of real world situations (7-1.7) and summarize their predictions using correct and clearly written or spoken words to communicate their understanding (7-1.6).

http://www.nctm.org/profdev/content.aspx?id=11688�


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Indicator 7-6.2

Organize data in box plots or circle graphs as appropriate.


In fifth grade, students applied procedures to calculate the measures of central tendency [median] (5-6.3). Fifth grade students used a protractor to measure angles from 0 to 180 degrees (5-5.2). In sixth grade, students analyzed which measure of central tendency is the most appropriate for a given purpose (6-6.3) and organized data in frequency tables, histograms, or stem-and-leaf plots as appropriate (6-6.2).

In seventh grade, students will organize data in box-and-whisker plots as well as in circle graphs (7-6.2).

In eighth grade, students interpret graphic and tabular data representations by using range and the measures of central tendency (8-6.8). They will also organize data in matrices or scatterplots as appropriate (8-6.2).

Taxonomy Level

Cognitive Dimension: Analyze Knowledge Dimension: Conceptual Key Concepts Vocabulary

• Box and whisker plot • Five Number Summary • Median • Lower Extreme • Upper Extreme • Lower Quartile • Upper Quartile • Outlier • Circle Graph • Central Angle • Degrees • Percents


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• Understand the purpose of box plots and circle graphs • Determine the Five Number Summary (median, lower and upper extremes,

lower and upper quartiles) • Recognize an outlier marked by an * on a box and whisker plot summary. • Understand that 100% of a circle is 360°. • Use proportions to determine the angle size of each central angle based on a

percent. • Utilize a protractor to measure angles less than or greater than 180°. • Understand equivalent ratios. • Develop the steps in constructing a circle graph as well as carry out the steps

(Write each category as a fraction and write each fraction as a percent. Express each category as a decimal, as a percent, and determine the degrees of each central angle.)

• Use proportions to solve for the unknown. For this indicator, it is not essential for students to:

• Determine the mean of the data sets. • Calculate the parameters of an outlier.


• The most common error when organizing into a box and whisker plots consists of failing to place a data set in least to greatest order prior to determining the five number summary.

• The most common error when organizing data into a circle graph is using the percent calculations as the degrees of central angles.

• Students may not understand that each quartile refers to one-fourth of the number of items, not one-fourth of the range.


• Human Box and Whisker Plot: http://www.learnnc.org/lp/pages/3767 - Students will learn how to construct box and whisker plots as they actively participate in being a part of one based upon their heights. As an extension of the lesson, students will learn how to interpret a graph of this type.

http://www.learnnc.org/lp/pages/3767�


5

Assessment Guidelines The objective of this indicator is to organize, which is in the “analyze conceptual” knowledge cell of the Revised Taxonomy. To organize is to determine how elements fit or function within a structure. The learning progression to organize requires students to generalize mathematical statements (7-1.5) about the purposes for using box and whisker plots and circle graphs (7-1.6). They determine a five number summary in order to analyze or create a box and whisker plot. They also determine how the parts (percents and angle degrees) of a circle graph relate to the whole in order to create a circle graph from data. Students should be able to describe the procedures to organize data using correct and clearly spoken words and mathematical notation (7-1.6). They should use deductive reasoning to reach a conclusion from known facts (7-1.5).


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Indicator 7-6.3

Apply procedures to calculate the interquartile range.


In third grade students first applied a procedure to find the range of a data set (3-6.1).

In seventh grade students should acquire the ability to calculate and interpret the interquartile range (7-6.3) and organize and interpret box and whisker plots (7-6.2).

In eighth grade students will interpret graphic and tabular data representations by using range and the measures of central tendency (8-6.8).

Taxonomy Level

Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts Vocabulary

• Interquartile range • Box-and-Whisker Plot • Five Number Summary • Median • Upper and Lower Extremes • Upper and Lower Quartiles


• Understand the meaning of interquartile range • Understand that quartiles are three numbers that divide an ordered set of

data into four equal sized subsets. • Understand that 25% of the data falls between two successive quartiles • Determine the median of a set of data. • Interpret the upper and lower quartiles of a set of data or from a box and

whisker plot


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• Use the quartiles to calculate the interquartile range. For this indicator, it is not essential for students to:

• Make a box-and-whisker plot Student Misconceptions/Errors

• Students may calculate the range as the interquartile range. • Intequartile range may be unusually difficult for some students to grasp as it

is a much higher order and more abstract concept. • Students may not understand the number of data elements between

successive quartiles depends on the size of the data set but the percent of the data between the successive quartiles is always 25%.

Instructional Resources and Stategies

• Box and Whisker Plot: http://mathbits.com/MathBits/TISection/Statistics1/BoxWhisker.htm - Use TI-83+ with box and whisker plots

• Location, Location, Location: Students will utilize simple grid paper strips to interpret or calculate the median, quartiles and interquartile range of data sets:http://public.doe.k12.ga.us/DWPreview.aspx?WID=89&obj=101363&mode=1

Assessment Guidelines The objective of this indicator is to apply which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Procedural knowledge is based on criteria for using skills, algorithms, techniques, and methods in familiar or unfamiliar tasks. The learning progression to apply requires students to understand the meaning and purpose of interquartile range. They use deductive reasoning (7-1.5) to demonstrate an understanding of the relationships between the upper and lower quartiles. Students must use correct and clearly written or spoken words and notation to communicate their understanding of how to calculate the interquartile range of a pair of box and whisker plots from data (7-1.6).

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Indicator 7-6.4

Interpret the interquartile range for data.


In third grade, students applied a procedure to find the range of a data set (3-6.1)

In seventh grade students interpret the interquartile range for data (7-6.4) and apply procedures to calculate the interquartile range (7-6.3).

In eighth grade, students interpret graphic and tabular data representations by using range and the measures of central tendency (8-6.8).

Taxonomy Level


• Interquartile range • Box-and-Whisker Plot • Five Number Summary • Median • Upper and Lower Quartiles


• Understand the meaning of interquartile range • Use appropriate mathematical language to interpret the meaning of the

interquartile range in real world situations


• Make a box-and-whisker plot


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• Students may determine range as the interquartile range. • Intequartile range may be unusually difficult for some students to grasp as it

is a much higher order and more abstract concept.

Instructional Resources

• Five Number Summary: http://illuminations.nctm.org/Lessons/States/FiveNumSum.htm – NCTM Illuminations

Assessment Guidelines The objective of this indicator is to interpret, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To interpret involves changing from one form of representation to another (e.g. numerical to verbal). The learning progression to interpret requires students to recall and understand the meaning of interquartile range. Students use their understanding analyze data in a variety of forms. They use inductive and deductive reasoning (7-1.5) to reach a conclusion and explain how the interquartile range impacts the data. Students explain and justify their answers using correct and clearly written or spoken words (7-1.6).

http://illuminations.nctm.org/Lessons/States/FiveNumSum.htm�


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Indicator 7-6.5

Apply procedures to calculate the probability of mutually exclusive, simple, or compound events.


In sixth grade, students use theoretical probability to determine the sample space and probability for one and two stage events (6-6.4) and applied procedures to calculate the probability of complementary events (6-6.5).

In seventh grade students will apply procedures to calculate the probability of mutually exclusive simple or compound events (7-6.5).

In eighth grade, students use theoretical and experimental probability to make inferences and convincing arguments about an event or events (8-6.3). They also apply procedures to calculate the probability of two dependent events (8-6.4) as well as interpret the probability for two dependent events (8-6.5)

Taxonomy Level

Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts Vocabulary

• Probability • Complementary Events • Complement • Mutually Exclusive • Not Mutually Exclusive • Population • Sample • Sample space • Possible outcomes



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• Add and subtract fractions with like denominators when determining probability of the complements of events.

• Find the probabilities of a simple event and its complement and describe the relationship between the two.

• Interpret probability notation. Ex. P(head or tails) when tossing a coin. • Multiply fractions when determining the probability of simple compound

events. • Explain that the word “or” in a probability situation indicates addition. • Determine that the sum of the probabilities of all the outcomes in a sample

space is 1; therefore, the sum of the complementary events is 1. • Multiply fractions when determining the probability of simple compound

events. • Explain that the word “then” in a probability situation indicates multiplication.


• Calculate the probability of dependent events (8th grade). Student Misconceptions/Errors

• Students may tend to misinterpret problem situations involving the use of the word “NOT” when determining the probability of the complement of an event.

• One misconception students have revolves around the concepts of sample space, sample size, compound events and simple events. By not having an innate understanding of sample space, a student may falsely assign similar probabilities to two different events.

Instructional Resources and Stategies

• Modeling with concrete objects such as spinners, cards, or marbles in a bag needs to be done in order for the students to develop a mental picture of the probabilities of complementary events.

What is the complement of:

P ( or ) using spinner 1?

Spinner 1

Spinner 2 Find the compound probability:

P ( then ) using spinner 1 and two?


12

• Probability – http://nlvm.usu.edu/en/nav/category_g_3_t_5.html Spinners - National Library of Virtual Manipulatives

Assessment Guidelines The objective of this indicator is to apply which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Procedural knowledge may be applied to a familiar task or to an unfamiliar task; therefore, the student’s procedural knowledge of probability should include various “real-world” type situations (7-1.7) including but not limited to the use or the simulation of the use of dice, cards, coins, etc. The learning progression to apply requires students to understand the meaning of mutually exclusive, simple and compound. They calculate the probability of the complement of an event and compound events in a real world context . They generalize connections (7-1.7) with these real world situations by using deductive reasoning to interpret the meaning of their answers. Students use correct and clearly spoken words and mathematical notation (7-1.6) to communicate their understanding. Students engage in ample opportunities to calculate, evaluate, and interpret the probability of mutually exclusive simple or compound (combination of at least two simple events) events (7-1.2).

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Indicator 7-6.6

Interpret the probability of mutually exclusive, simple, or compound events.


In sixth grade, students use theoretical probability to determine the sample space and probability for one and two stage events (6-6.4) and applied procedures to calculate the probability of complementary events (6-6.5).



Taxonomy Level


• Probability • Complementary Events • Complement • Mutually Exclusive • Not Mutually Exclusive • Population • Sample • Sample space • Possible outcomes • Compound Events • Independent Events



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• Interpret probability notation. Ex. P(head or tails) when tossing a coin. • Interpret that the word “or” in a probability situation indicates addition. • Determine that the sum of the probabilities of all the outcomes in a sample

space is 1; therefore, the sum of the complementary events is 1. • Interpret that the word “then” in a probability situation indicates

multiplication.


• Interpret the probability of dependent events (8th grade). Student Misconceptions/Errors

• Students may tend to misinterpret problem situations involving the use of the word “NOT” when determining the probability of the complement of an event.

• One misconception students have revolves around the concepts of sample space, sample size, compound events and simple events. By not having an innate understanding of sample space, a student may falsely assign similar probabilities to two different events.


• Modeling with concrete objects such as spinners, cards, or marbles in a bag needs to be done in order for the students to develop a mental picture of the probabilities of complementary events.

What is the compound probability:

P ( then ) using spinner 1 twice?

Spinner 1

Spinner 2 Find the compound probability:

P ( then ) using spinner 1 and 2?


15

Assessment Guidelines The objective of this indicator is to interpret, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To interpret is to change from one form (numerical) to another form (verbal). The learning progression to interpret requires students to understand the meaning of mutually exclusive, simple or compound events. Given a real world problem and a probability, students use inductive and deductive reasoning (7-1.3) and summarize observations. They use their understanding of the characteristics of these types of events to generalize mathematical statements (7-1.5) about their observations. Students use correct and clearly written or spoken words to communicate their understanding.


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Indicator 7-6.7

Differentiate between experimental and theoretical probability of the same event.


In sixth grade, students used theoretical probability to determine the sample space and probability for one- and two-stage events using tree diagrams, models, lists, charts, and pictures (6-6.4).

In seventh grade students will find and differentiate between the theoretical and experimental probability of the same events. (7-6.7)

In eighth grade students will use theoretical and experimental probability to make inferences and convincing arguments about an event or events (8-6.3)

Taxonomy Level

Cognitive Dimension: Analyze Knowledge Dimension: Factual Key Concepts Vocabulary

• Experimental Probability • Theoretical Probability • Simulation


• Understand the characteristics of theoretical probability • Understand the characteristics of experimental probability • Understand the purpose of a simulation • Understand that experimental probability is the background for examining

theoretical probability • Understand that the experimental approach has its limitations because it is

impossible to perform an infinite number of trials • Understand the relationship between a simulation and experimental

probability


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• Understand that when all outcomes of an experiment are equally likely, the theoretical probability of an event is the fraction of the outcomes in which the event occurs


• Find the probability of compound events. Student Misconceptions/Errors

• Students may confuse what has happened with what may happen; • Students may believe that because an event has recently happened it has a

high probability of reoccurring; • Students may use too small a sample size when using experimental relative

observed frequencies as a measure of probability • Students may make the assumption that outcomes are equally likely (in order

to calculate theoretical probabilities) when they are not. • Students may assume that the theoretical probability and observed relative

frequency of an event will be the same • Students may allot a probability greater than one to an event. • Students may fail to systematically list possible outcomes and thereby miss

counting some outcomes or count some outcomes more than once. • Students may fail to understand that when all outcomes of an experiment are

equally likely, the theoretical probability of an event is the fraction of the outcomes in which the events occurs.

Instructional Resources and Strategies In order for students to develop an understanding of the concepts of theoretical and experimental probabilities, instructional strategies should elaborate those concepts. Students should understand that a simulation is a procedure for answering questions about a real problem by conducting an experiment that closely resembles the real situation. As students think about and discuss the real problem and the factors that make the real problem more complex, they should be reminded that a simulation is an approximation of the real problem and that with a simulation some of the factors are eliminated such as any possible danger, complexity of the problem, or length of time necessary to solve the problem.


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• Modeling with concrete objects such as spinners, dice, coins, cards, or marbles in a bag needs to be done in order for the students to develop a mental picture of the probabilities of experimental events as compared to the theoretical probability.

• Quick-Take on ... Theoretical and Experimental Probability: http://msteacher.org/epubs/math/QuickTakes/exp_probability.aspx Gives middle school students opportunities to think about probability to understand the meaning of taking a chance.

• Monty Hall, Let’s Make a Deal: http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html - This applet can be used to help introduce the experimental probability than the other remaining door. Have each student commit to an answer. Tell them that by the end of class they will know enough to solve the problem. For a concrete example, provide each pair of students with cups and a small item (such as a coin, button, eraser) to simulate the Monty Hall problem. Combine data from each pair to get the sum of the classes’ data on a spreadsheet.


The objective of this indicator is to differentiate which is in the “analyze procedural” knowledge cell of the Revised Taxonomy. To differentiate is to distinguish relevant from irrelevant parts or important from unimportant parts of presented material. The learning progression to differentiate requires students to recall the meaning of experimental and theoretical probability. Students explore these concepts through real world applications in order to generalize connections between probability and real world situations (8-1.7). Based on their observations, they generalize mathematical statements using deductive and inductive reasoning (8-1.5). They understand the role of simulations and use correct and clearly written or spoken words to communicate their understanding (8-1.6). They use their understanding to distinguish similarities and difference between the two types and probability.

http://msteacher.org/epubs/math/QuickTakes/exp_probability.aspx�

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Standard 7-6: The student will demonstrate through the mathematical processes an understanding of the relationships between two populations or samples. Indicator 7-6.8 Use the fundamental counting principle to determine the number of possible outcomes for a multistage event Continuum of Knowledge In sixth grade, students use theoretical probability to determine the sample space and probability for one and two stage events (6-6.4) and applied procedures to calculate the probability of complementary events (6-6.5).



Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts

• Fundamental Counting Principle • Outcomes • Multi-stage event


• Understand the Fundamental Counting principle • Use a method for recording outcomes such as lists, tree diagram, etc.. • Understand that it does not matter with which event they begin, the number

of outcomes is the same because of the commutative property of multiplication


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For this indicator, it is not essential for students to: None noted Student Misconceptions/Errors Students may not understand the number of outcomes will be the same regardless of which event they begin with. Instructional Resources and Strategies If ice cream comes in either a cup or a cone and comes in three different flavors, what are the possible outcomes? (Six) / chocolate cup / chocolate < / \ chocolate cone / / / strawberry cup <-- strawberry < \ \ strawberry cone \ \ / vanilla cup \ vanilla <

\ vanilla cone But if students start with a cup or cone, there will still be six outcomes. / chocolate cup / / cup <-- strawberry cup / \ / \ vanilla cup / < \ \ / chocolate cone \ / \ cone <-- strawberry cone \ \ vanilla cone

Assessment Guidelines The objective of this indicator is to use which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Procedural knowledge is the knowledge of steps and the criteria of when to use those steps. The learning progression to use requires students to recall the meaning of outcomes and how to construct tree diagrams, lists, etc… Students use their prior knowledge of single and two stage events to


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generalize connections (7-1.7) with multistage events. They generate and solve complex problems (7-1.1) and explain and justify their answers using correct and clearly written or spoken words (7-1.6).

Documents

Standard 7-2: The student will demonstrate through the ...toolboxforteachers.s3.amazonaws.com/.../7th...Docs.pdf · relationships between fractions and fractional percents, mixed