Guidelines

Format of Guidelines

STRAND

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM

SOLVING EXAMPLE

CONNECTIONS, INTEGRATION

RESOURCES

TOPIC

Description of the extent to which a skill is developed, or of what behaviors the students will exhibit in completing the activities used to develop the skill.

Vocabulary with which the students should be able to communicate (speak, write) mathematically.

Computational or conceptual representation of skill indicating the extent to which the skill is to be presented at this level.

Application of skill to problem-solving/real-life situation.

Inter-relatedness of skill to real-life or applications of skill to other areas of the curriculum or to other skills in mathematics.

This space is intended to be utilized by the teacher to list the references and materials appropriate to the development of the skill: e.g. teacher guide, technology, supplementary materials, manipulatives, etc.

STRAND refers to the general area of mathematics into which each topic is categorized. Strands include READINESS SKILLS, NUMBER CONCEPTS, OPERATIONS, PROBLEM SOLVING, MENTAL MATHEMATICS, ESTIMATION, GEOMETRY, MEASUREMENT, REASONINGS SKILLS, DATA NA DSTATISTICS, TECHNOLOGH, RATIO-PROPORTION-PERCENT, AND PRE-ALGEBRA SKILLS.

A section of these guidelines is also designated as ENRICHMENT. This is intended for the horizontal expansion of topics rather than the vertical progression of skills.

TOPIC – specific area within each strand. (For example, USING A PROBLEM-SOLVING MODEL is a topic in the strand of PROBLEM SOLVING).

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

NUMBER SENSE

M-6.1: The students will understand place value of decimals and whole numbers millionths through billions.

millionths

hundred-thousandths

ten-thousandths

thousandths

hundredths

tenths

units

tens

hundreds

thousands

ten-thousands

hundred-thousands

millions

ten-millions

hundred millions

billions

numeral

digit

7,824,315 reads seven million, eight hundred twenty four thousand, three hundred fifteen.

Caution: Do not allow the students to read the word “and” when reading whole numbers. “And” indicates the placement of a decimal point. (e.g. 8.4 is “eight and four tenths.”

Draw a place value chart for forty nine billion

Show forty nine billionths on a place value chart.

Write a ten-digit number with a six in the millionths place and a four in the hundred-thousands place.

Study Skills: Use of reference materials.

Language Arts: Writing large numbers within a paragraph/sentence.

Social Studies, Science: larger and smaller numbers used in data.

Physical Education: number used to record speed and distances.

Technology: limitations of calculator display.

NCTM Addenda Series: Number Sense

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

READING AND WRITING NUMBERS (NUMERALS)

M-6.2: The students will continue to develop skill in reading and writing numbers.

number

numeral

Reading: Stress use of comma as an aid in reading whole numbers and the use of the decimal point in separating whole numbers and decimals.

Writing: Stress correct formation of numerals and mathematical symbols; correct size of fractions and exponents.

Write in words:

4,003,016.024

(four million, three thousand, sixteen and twenty-four thousandths).

Write the standard numeral for:

Sixteen billion, twenty-two thousand, one hundred fifty three and twenty-one hundredths.

(16,000,022,153.21)

Language Arts: Handwriting Reading - vocabulary enrichment (number-related words).

Spelling – correct use of -th at the end of words indicating decimals: tenths, hundredths, etc.

Correct writing of numbers in sentences outlines, tables, etc.

Social Studies, Science: reading and writing numerical data.

Technology: interpreting

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

$Oval: · · ·$ $Oval: 1 2 1 4$

NUMBER SENSE

Fractions

M-6.3: The students will review the definitions of a fraction as

1) equal parts of a whole; or,

2) equal parts of a set.

fraction

fraction bar

numerator

denominator

· · ·

Tess cut the pizza into ten equal pieces and gave Monica two pieces. Draw a picture to show this. What fraction does Monica’s share represent?

Real Life: give the students practice dividing items into fractional parts.

$Text Box:$

NUMBER SENSE

Decimals

M-6.4: The students will understand the concept of decimals.

M.6.5 The students will be able to change decimals to fractions and percents.

decimal

decimal point

decimal fraction

decimal square

per cent

Use graph paper/decimal squares to illustrate concept of a decimal as part of the whole represented in units divisible by ten—tenths, hundredths, thousandths, etc.

The decimal square has 100 parts. Twenty-two of these are covered. This can be represented by the decimal numeral 0.22.

Real Life:

Money – a penny is one one-hundredth of a dollar; a dime is one-tenth of a dollar.

Time – a year is one-one-hundredth of a century; a year is one-tenth of a decade.

Language Arts:

Vocabulary – note words with dec– as a root – December, decade, decagon, etc.

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

NUMBER PATTERNS

M-6.6: The students will continue to discover and complete number patterns.

multiples

skip counting

patterns

number sequence

ordinal numbers

cardinal numbers

2, 5, 8, 11, 14

(+3)

1, 5, 7, 11, 13,17 (+4, then +2)

Create a pattern that adds 5, then 4, then 3, then 2, and so on to get to succeeding numbers in the pattern.

Extend to include fractions and decimals in patterns.

Science: patterns in nature

Social Studies: Olympic years

Real life: pattern in architecture, industry, sports

Math: Geometry – using pattern blocks or cubes, have the students discover patterns with a rectangular or triangular arrangement.

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

READING AND WRITING

Mathematical Symbols

M-6.7: The students will recognize, read and write mathematical symbols.

M-6.8: The students will recognize the fraction bar as a symbol of division.

symbol

+, —, %, =, ¸, ), <, >, [, ³, g, l, …,

Ð3°, %, p, 2/3

fraction bar

decimal point

bar notation for repeating decimals

Provide practice using symbols in context.

2/3 > 3/5

0.4 = 0.444....

25%

mÐABC=90°

45)9225

Write an equation or inequality to show:

Miss Kane’s class completed five-tenths of their assignments by noon. Miss Butler’s class completed six-tenths of their work at the same time.

(0.6 > 0.5)

5,436 attended an Olympic event. About 30% were children. About how many were children?

Real Life: Reading signs

Language Arts: Using abbreviations

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

COMPARING AND ORDERING NUMBERS

Whole numbers

Fractions

Decimals

M-6.9: Given real-life situations, the students will compare numbers

greater than

less than

equal

equivalent

1. Compare on number line: number to the right is always the greater number.

2. Use place-value chart to compare whole numbers.

3. Using graph paper, fraction bars, or decimal squares, create models of numbers to compare.

Items to compare:

sports’ statistics

gas mileage

nutritional value of food

prices

Extend practice to include the comparison of fractions with decimals:

0.3……..1/3

Real Life: Comparison shopping

Social Studies: distances on maps

Study Skills: Reading charts, tables, etc.

Math: problem solving, measurement

NUMBER LINE

Ordering numbers from greatest to least and vice versa

M-6.10: The students will use a number line as an aid in ordering whole numbers, decimals, and fractions.

M-6.11: Given a set of data, the students will order the data from least to greatest or from greatest to least.

number line

Order the following from least to greatest:

0.2, 1.0, 1.2, 0.75

3/5, 1/2, 0.4

In the long jump, Harry jumped 17.2 feet; Jake jumped 16.9 feet and Bill jumped 15.85 feet. Show their placement on a number line.

Language Arts: Write a step-by-step plan for comparing numbers.

Technology: Utilize software programs which compare numbers.

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

ROUNDING WHOLE NUMBERS AND DECIMALS

M-6.12: The students will round whole numbers and decimals to any place-value position.

rounding

Round 53,769.217 to the nearest hundredth.

(The digit one is in hundredths place. The number immediately to the right is 7 which is greater than 5, so 53,769.217 rounded to the nearest hundredth is 53,769.22

Locate sizes of countries in an atlas or almanac. Round these numbers to the nearest thousand. Order these from greatest to least.

Mathematics: measurement, estimating

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

EXPANDED NOTATION

M-6.13: The students will express number through billions using expanded notation.

standard form

exponential notation

805,602,000,000

800,000,000,000

+ 5,000,000,000

+ 600,000,000

+ 2,000,000

Write the numeral 745,274,103,204

Using expanded notation.

Study Skills: Outlining skills, finding smaller components of the whole number.

POWERS AND EXPONENTS

M-6.14: The students will understand that when a number is multiplied by itself it can be expressed as a base with an exponent.

factors

exponent

base

powers

exponent

100,000 = 10⁵

base

Write 2³ as a product.

(2³ = 2 x 2 x 2)

3 factors

Write 5x5x5x5 using exponents.

(5⁴)

Which of the following statements is correct:

A. 3² = 2³

B. 4² = 2⁴

B is correct.

A. 9 ¹ 8

B. 16 = 16

Express a billion as a power of ten.

(10⁹)

Science and Social Studies: Use of larger numbers.

D

ESTIMATING

Estimation skills are developed in various places throughout these guidelines.

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

M

FACTORS

Common factors

M-6.15: The students will identify common factors of a set of numbers.

factors

common factors

greatest common factors

Find the common factors of 12 and 16.

Factors of 12 –

1, 2, 3, 4, 6, 12

Factors of 16 –

1, 2, 4, 8, 16

1, 2 and 4 are common factors.

4 is the greatest common factor.

One group of chairs has 16. Another group ahs 12. The chairs are arranged in rows of the same number. What is the greatest amount that could be in each row?

Mathematics: patterns, problem solving

Real Life: Careers in art and design.

MULTIPLES

Common

Multiples

Least Common Multiple

M-6.16: The students will identify common multiples of a set of numbers; they will determine the least common multiple.

multiple

common multiple

least common multiple

Multiples of 8: 8, 16, 24, 32, 40, 42……

Multiples of 12:

12, 24, 36, 48….

Common multiples of 8 and 12 are 24 and 48. The least common multiple is 24.

What is the smallest number of chairs that would fit into rows of 12 or rows of 8 without having any left over?

Mathematics: least common denominator of a fraction; problem solving – make a model, draw a picture strategies.

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

PRIMES AND COMPOSITES

M-6.17: The students will become familiar with the concepts of prime and composite numbers.

prime

composite

A prime number is a number that has 2 factors – itself and one.

2, 3, 5, 7, and 11 are prime.

Composite numbers have more than two factors.

4, 6, 8, 9, and 10 are composite numbers.

NOTE: 0 and 1 are neither prime nor composite.

Find the prime numbers that fall between 1 and 40.

Explain why each of the following is either prime or composite.

45, 36, 17, 23, 41

How many different ways can Sue arrange 31 apples on a tray if she wants to have the same amount apples in each row? 36 apples?

Mathematics: patterns, problem solving.

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

RATIOS

M-6.18: The students will express a fraction as a ratio of two numbers.

M-6.19: The students will identify ratios and will write equivalent ratios

ratio

rate

2/7 can be expressed as “2 out of 7”

2 to 7

2:7

Express as a ratio: the cake had eight slices. Four were eaten.

Paul saves $0.75 a week. John saves $1.25. What is the ratio of John’s savings to Paul’s?

Language Arts: Explore commercial ads which claim that 9 out of 10 people buy…..

Mathematics: Graphs

D

PROPORTION

M-6.20: The students will understand the concept of proportion and will identify the terms of a proportion.

M-6.21: The students will express proportions in terms of equivalent fractions.

proportion

equivalent fractions

equivalent ratios

means

extremes

cross-product multiplication

2 ₌6

3 9

Solve for n:

3 ₌ n

4 36

Explain: The product of the means is equal to the product of the extremes.

If 3 buses hold a total of 74 people, what is the ratio of people to the buses in lowest terms?

If four pounds of peaches cost $1.96, what is the cost of one pound?

Mathematics: Problem solving – act it out, make a model, draw a picture strategies.

RECIPROCALS

M-6.22: The students will identify the reciprocal or multiplicative inverse

reciprocal

multiplicative inverse

The reciprocal of 3/4 is 4/3 because

3 _x 4 ₌ 1

4 3

Mathematics: Inverse operations.

Language Arts: Antonyms

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

BASES OTHER THAN TEN

M-6.23: The students will become familiar with bases other than base ten.

bases

base ten

decimal system

base two

binary system

Review the meaning of base ten:

123 means:

1 x 10^{2 +}

2 x 10^{1 +}

3 x 10⁰

= 100 + 20 + 3

= 123

Note: The only digits used in a number base are the digits less than the base itself:

Base two – 0, 1

Base 3 – 0, 1, 2

Base 4 – 0, 1, 2, 3

Technology: Use of binary numbers in computer technology.

ROMAN NUMERAL

M-6.24: The students will understand the structure of Roman numerals and will apply their use in other areas.

I = 1

V = 5

X = 10

L = 50

C = 100

D = 500

M = 1000

Lines connecting the Roman numerals indicate multiplication by 1000. Students should be instructed that they should not draw lines connecting the letters at top and bottom.

Explain: a Roman numeral of lesser value preceding one of greater value indicates subtraction.

eg. XL means 50 – 10 or 40

A Roman numeral of lesser value following one of greater value indicates addition.

CIII = 103

Write the following equations using Roman numerals:

34 x 46 =

69 + 146 =

212 – 116 =

Real Life: Clocks, outlines copyright dates, dates in film credits building cornerstones.

READINESS CONCEPTS, NUMBER CONCEPTS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

D

INTEGERS

M-6.25 The students will become familiar with the concept of positive and negative numbers.

integer

positive

negative

Use in real life context – degrees below zero, feet below sea level, yards lost on a football play, etc.

Represent on a number line.

Write a number to show:

eighteen degrees below zero;

a loss of 20 points in a card game.

Real life: Point out the use of positive and negative numbers in calculating yards lost and gained in a football game.

OPERATIONS – GRADE 6

SEQUENCE

SKILL

OBJECTIVE

VOCABULARY

EXAMPLE

PROBLEM SOLVING EXAMPLE

CONNECTIONS,

INTEGRATION

RESOURCES

USE CONCETE OBJECTS TO MODEL OPERATIONS

M-6.26: Manipulative materials – connecting cubes, blocks, centimeter cubes, beans, etc. should be used frequently when modeling concepts.

M-6.27: Understanding is built when skills are presented following the concept continuum.

OPERATIONS INTRODUCED AND DEVELOPED IN PROBLEM-SOLVING CONTEXT

M-6.28: When operations are presented, it is important to give purpose to the learning by presenting the material in a problem – solving/real life context. Equations should never be presented in isolation until mastery has occurred.

M-6.29: Extend to include two and three step problems and using a single equation to solve.

e.g. The chef had 4 three-pound bags of flour and 2 five-pound bags of flour. He used six pounds of flour in

today’s recipes. How much flour was left?

(4 x 3) + (2 x 5) – 6 =

USE SYMBOLS FOR OPERATIONS

M-6.30: Refer to “symbols” in Readiness and Number Concepts section of these guidelines.

M

USE SINGLE– DIGIT AND MULTI – DIGIT NUMBERS

M-6.31: Review and maintain material presented at earlier levels Extend through millions. Students should make frequent use of the calculator to check material which they have already mastered.