Students investigate magnitude with positive and negative numbers. Number outcome 1Mastery Cove ref
Students investigate magnitude with positive and negative numbers.
Guiding question
- How can the infinite nature of the number line broaden the perception of number?
Knowledge
- Negative numbers are to the left of zero on the number line visualized horizontally, and below zero on the number line visualized vertically.
- Positive numbers can be represented symbolically with or without a positive sign (+).
- Negative numbers are represented symbolically with a negative sign (−).
- Zero is neither positive nor negative.
- Negative numbers communicate meaning in context, including
- temperature
- debt
- elevation
- Magnitude is a number of units counted or measured from zero on the number line.
- Every positive number has an opposite negative number with the same magnitude.
- A number and its opposite are called additive inverses.
- The set of integers includes all natural numbers and their additive inverses.
- The sum of any number and its additive inverse is zero.
- The sum of two positive numbers is a positive number.
- The sum of two negative numbers is a negative number.
- The sum of a positive number and a negative number can be interpreted as the sum of zero and another number.
- Subtracting a number is the same as adding its additive inverse.
Understanding
- Symmetry of the number line extends infinitely to the left and right of zero or above and below zero.
- Direction relative to zero is indicated symbolically with a positive or negative sign.
- Magnitude with direction distinguishes between positive and negative numbers.
- Any number can be expressed as a sum in infinitely many ways.
- The difference of any two numbers can be interpreted as a sum.
Skills & procedures
- Identify negative numbers in familiar contexts, including contexts that use vertical or horizontal models of the number line.
- Express positive and negative numbers symbolically, in context.
- Relate magnitude to the distance from zero on the number line.
- Relate positive and negative numbers, including additive inverses, to their positions on horizontal and vertical models of the number line.
- Compare and order positive and negative numbers.
- Express the relationship between two numbers, including positive and negative numbers, using < , > , or = .
- Investigate addition of an integer and its additive inverse.
- Express zero as the sum of integers in multiple ways.
- Model the sum of two positive integers.
- Model the sum of two negative integers.
- Model the sum of a positive and negative integer as the sum of zero and another integer.
- Express a difference as a sum.
Source
Practice coming soon
Students solve problems, using standard algorithms for addition and subtraction. Number outcome 2Mastery Cove ref
Students solve problems, using standard algorithms for addition and subtraction.
Guiding question
- How can the processes of addition and subtraction be applied to problem solving?
Knowledge
- Standard algorithms are reliable procedures for addition and subtraction.
- Contexts for problems involving addition and subtraction include money and metric measurement.
Understanding
- Addition and subtraction of numbers in problem-solving contexts is facilitated by standard algorithms.
Skills & procedures
- Solve problems in various contexts, using standard algorithms for addition and subtraction.
Source
Practice coming soon
Students analyze numbers, using prime factorization and exponentiation. Number outcome 3Mastery Cove ref
Students analyze numbers, using prime factorization and exponentiation.
Guiding question
- How can prime factorization and exponentiation provide new perspectives of numbers?
Knowledge
- The order in which three or more numbers are multiplied does not affect the product (associative property).
- Any composite number can be factored and expressed as a product of smaller numbers.
- Prime factorization represents a number as a product of prime numbers.
- Any composite factor of a number can be determined from its prime factors.
- Repeated multiplication of identical factors can be represented symbolically as a power (exponentiation).
- A power, A^n , includes a base, A , representing the repeated factor, and an exponent, n , indicating the number of repeated factors.
- Any repeated prime factor within a prime factorization can be expressed as a power.
Understanding
- A product can be composed in multiple ways.
- The prime factors of a number provide a picture of its divisibility.
- Different representations of a product can provide new perspectives of its divisibility.
- A power is divisible by its base.
Skills & procedures
- Compose a product in multiple ways, including with more than two factors.
- Express the prime factorization of a composite number.
- Determine common factors for two natural numbers, using prime factorization.
- Determine divisibility of a natural number from its prime factorization.
- Identify the base and exponent in a power.
- Express the product of identical factors as a power, including within a prime factorization.
- Describe the divisibility of numbers represented in various forms.
Source
Practice coming soon
Students apply standard algorithms to multiplication and division of decimal and natural numbers. Number outcome 4Mastery Cove ref
Students apply standard algorithms to multiplication and division of decimal and natural numbers.
Guiding question
- How can the processes of multiplication and division be applied to decimal numbers?
Knowledge
- Standard algorithms are reliable procedures for multiplication and division of numbers, including decimal numbers.
- A quotient with a remainder can be expressed as a decimal number.
Understanding
- Multiplication and division of decimal numbers is facilitated by standard algorithms.
Skills & procedures
- Explain the standard algorithms for multiplication and division of decimal numbers.
- Multiply and divide up to 3-digit natural or decimal numbers by 2-digit natural numbers, using standard algorithms.
- Assess the reasonableness of a product or quotient, using estimation.
- Solve problems, using multiplication and division, including problems involving money.
Source
Practice coming soon
Students relate fractions to quotients. Number outcome 5Mastery Cove ref
Students relate fractions to quotients.
Guiding question
- How can equal sharing contribute meaning to fractions?
Knowledge
- An equal-sharing situation can be represented by a fraction in which the numerator represents the quantity to be shared and the denominator represents the number of shares.
- Division can be used to determine an equal share.
- Division of the numerator by the denominator of a fraction provides the equivalent decimal number.
Understanding
- Fractions represent quotients in equal-sharing situations.
- All equivalent fractions represent the same quotient.
Skills & procedures
- Model an equal-sharing situation in more than one way.
- Describe an equal-sharing situation, using a fraction.
- Express a fraction as a division statement and vice versa.
- Convert a quotient from fraction to decimal form, using division.
Source
Practice coming soon
Students add and subtract fractions with denominators within 100. Number outcome 6Mastery Cove ref
Students add and subtract fractions with denominators within 100.
Guiding question
- How can the addition and subtraction of fractions be generalized?
Knowledge
- Addition and subtraction of fractions is facilitated by representing the fractions with common denominators.
- Denominators are related if one is a multiple of the other.
- Multiplication of one denominator by the factor that relates it to another denominator achieves common denominators.
- The product of the denominators of two fractions provides a common denominator.
Understanding
- Fractions with common denominators have the same units.
- Any numbers with the same unit can be compared, added, or subtracted.
Skills & procedures
- Recognize two fractions with related denominators.
- Determine the factor that relates one denominator to another.
- Express two fractions with common denominators.
- Add and subtract fractions.
- Solve problems involving addition and subtraction of fractions.
Source
Practice coming soon
Students interpret the multiplication of natural numbers by fractions. Number outcome 7Mastery Cove ref
Students interpret the multiplication of natural numbers by fractions.
Guiding question
- How can an understanding of multiplication be extended to fractions?
Knowledge
- Multiplication of a natural number by a fraction is equivalent to multiplication by the fraction’s numerator and division by its denominator.
- \enspace \enspace \enspace \enspace \enspace a \times \dfrac{b}{c} = \dfrac{ab}{c}
- Multiplication by a unit fraction is equivalent to division by its denominator.
- \enspace \enspace \enspace \enspace \enspace a \times \dfrac{1}{b} = \dfrac{a}{b}
- The product of a fraction and a natural number is the fraction with
- a numerator that is the product of the numerator of the given fraction and the natural number
- a denominator that is the denominator of the given fraction
- \enspace \enspace \enspace \enspace \enspace \dfrac{a}{b} \times c = \dfrac{ac}{b}
Understanding
- Multiplication does not always result in a larger number.
- Multiplication of a natural number by a fraction can be interpreted as repeated addition of the fraction.
- Multiplication of a fraction by a natural number can be interpreted as taking part of a quantity.
Skills & procedures
- Relate multiplication of a natural number by a fraction to repeated addition of the fraction.
- Multiply a natural number by a fraction.
- Relate multiplication by a unit fraction to division.
- Multiply a natural number by a unit fraction.
- Model a fraction of a natural number.
- Multiply a fraction by a natural number.
- Solve problems, using multiplication of a fraction and a natural number.
Source
Practice coming soon
Students apply equivalence to the interpretation of ratios and rates. Number outcome 8Mastery Cove ref
Students apply equivalence to the interpretation of ratios and rates.
Guiding question
- In what ways can equivalent ratios support proportional reasoning?
Knowledge
- A proportional relationship exists when one quantity is a multiple of the other.
- Equivalent ratios can be created by multiplying or dividing both terms of a given ratio by the same number.
- A proportion is an expression of equivalence between two ratios.
- A rate describes the proportional relationship represented by a set of equivalent ratios.
- A unit rate expresses a proportional relationship as a rate with a second term of 1.
- A percentage describes a proportional relationship between a quantity and 100.
- Percent of a number can be determined by multiplying the number by the percent and dividing by 100.
Understanding
- All equivalent ratios express the same proportional relationship.
- A rate can be used to extend a given proportional relationship to different quantities.
Skills & procedures
- Determine whether two ratios are equivalent.
- Determine an equivalent ratio, using a proportion.
- Express a unit rate to represent a given rate, including unit price and speed.
- Relate percentage of a number to a proportion.
- Determine a percent of a number, limited to percentages within 100 \% .
- Solve problems involving ratios, rates, and proportions.
Source
Practice coming soon