MA19.4.28a
Describe right triangles as a category, and identify right triangles.
Describe right triangles as a category, and identify right triangles.
MA19.4.28a
MA19.4.29
Define a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts.
Define a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts.
Unpacked Content
Knowledge
Students know:
- Characteristics of lines of symmetry.
Skills
Students are able to:
- Define a line of symmetry for a two-dimensional figure.
- Identify and draw lines of symmetry for two-dimensional figures.
Understanding
Students understand that:
- a line of symmetry divides a shape into two parts such that when folded on the line, the two parts match.
Vocabulary
- Symmetry
- Two dimensional figure
- Line of symmetry
MA19.4.29a
Identify line-symmetric figures and draw lines of symmetry.
Identify line-symmetric figures and draw lines of symmetry.
MA19.5.1
Write, explain, and evaluate simple numerical expressions involving the four operations to solve up to two-step problems. Include expressions involving parentheses, brackets, or braces, using commutative, associative, and distributive properties.
Write, explain, and evaluate simple numerical expressions involving the four operations to solve up to two-step problems. Include expressions involving parentheses, brackets, or braces, using commutative, associative, and distributive properties.
Unpacked Content
Knowledge
Students know:
- Vocabulary associated with the four operations to write the symbolic notation of the mathematical expression.
Example: The phrase, "the product of 4 and 3" is written as "4 x 3." - Strategies for evaluating a numerical expression and replace it with an equivalent form.
Example: Given (22 + 16) + 43 can be replaced with 38 + 43 and then further simplified.
Skills
Students are able to:
- Write, explain, and evaluate numerical expressions representing two-step problems in context.
- Evaluate numerical expressions with grouping symbols.
- Translate a numerical expression into words.
- Write a numerical expression given a mathematical expression in words.
Understanding
Students understand that:
- multi-step word problems can be represented by numerical expressions using operations and grouping symbols to indicate order of evaluating them.
Vocabulary
- Expression
- Calculate
- Interpret
- Evaluate
- Compare
- Reasoning
- Grouping symbol
- Parentheses
- Braces
- Brackets
- Commutative property
- Associative property
- Distributive property
MA19.5.2
Generate two numerical patterns using two given rules and complete an input/output table for the data.
Generate two numerical patterns using two given rules and complete an input/output table for the data.
Unpacked Content
Knowledge
Students know:
- Strategies to identify numerical patterns and recognize the relationship between the terms in the pattern.
- Reasoning strategies to generate a numerical pattern which follow a given rule.
Skills
Students are able to:
- Generate two numerical patterns using two given rules.
- Complete an input/output table for data.
- Identify relationship between terms in an input/output table.
- Form ordered pairs from an input/output table.
- Graph ordered pairs on a coordinate plane.
Understanding
Students understand that:
- relationships between two numerical patterns can be represented by ordered pairs and graphed in the first quadrant of the coordinate plane.
Vocabulary
- Number pattern
- Graph
- Coordinate plane
- X-axis
- Y-axis
- Origin
- X-coordinate
- Y-coordinate
- Ordered pair
- Generate
- Sequence
MA19.5.2a
Use data from an input/output table to identify apparent relationships between corresponding terms.
Use data from an input/output table to identify apparent relationships between corresponding terms.
MA19.5.2b
Form ordered pairs from values in an input/output table.
Form ordered pairs from values in an input/output table.
MA19.5.2c
Graph ordered pairs from an input/output table on a coordinate plane.
Graph ordered pairs from an input/output table on a coordinate plane.
MA19.5.3
Using models and quantitative reasoning, explain that in a multi-digit number, including decimals, a digit in any place represents ten times what it represents in the place to its right and $\frac{1}{10}$ of what it represents in the place to its left.
Using models and quantitative reasoning, explain that in a multi-digit number, including decimals, a digit in any place represents ten times what it represents in the place to its right and $\frac{1}{10}$ of what it represents in the place to its left.
Unpacked Content
Knowledge
Students know:
- Each place value position represents 10 times what it represents in the place to its right.
Example: In 433, the underlined 3 represents 3 tens and has a value of 30 which is ten times the value of the 3 ones to its right. - Place value understanding is extended to apply reasoning that a place value position represents 1/10 of what it represents in the place value to its left.
Example: In 433, the underlined 3 represents 3 ones and has a value of 3 which is one-tenth of the value of the 3 tens or 30 to its left. - A given number multiplied by a power of 10 shifts the digits in the given number one place value greater (to the left) for each factor of 10.-A given number divided by a power of 10 shifts the digits in the given number one-tenth of the value (to the right) for each factor of 10.
Skills
Students are able to:
- Reason and explain the relationship between two successive place values.
- Explain patterns of zeros of the product when multiplying by powers of 10.
- Explain patterns in placement of decimals when multiplying or dividing by power of 10.
- Write powers of 10 using exponential notation.
Understanding
Students understand that:
- The relationship of adjacent places values in the base ten system extend beyond whole numbers to decimal values.
- Multiplying or dividing by a power of 10 shifts the digits in a whole number or decimal that many places to the left or right respectively.
Vocabulary
- Digit
- Decimal
- Decimal point
- Thousandths
- Hundredths
- Tenths
- Base-ten
- Expanded form
- Place value
- Power of 10
- Factor
- Base
- Exponent
- Product
MA19.5.3a
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, using whole-number exponents to denote powers of 10.
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, using whole-number exponents to denote powers of 10.
MA19.5.3b
Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10, using whole-number exponents to denote powers of 10.
Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10, using whole-number exponents to denote powers of 10.
MA19.5.4
Read, write, and compare decimals to thousandths.
Read, write, and compare decimals to thousandths.
Unpacked Content
Knowledge
Students know:
- How to read and write whole numbers in standard form, word form, and expanded form.
- How to compare two whole numbers using place value understanding.
- Prior place value understanding with whole numbers is extended and applied to decimal values.
- Recognize and model decimal place value using visual representations to compare.
Skills
Students are able to:
- Read and write decimal values in word form, standard form, and expanded form.
- Compare decimals to thousandths using , or = .
Understanding
Students understand that:
- the adjacent place value relationship in the base ten system extends to decimals and is used to write decimals in expanded form and compare decimals.
Vocabulary
- Compare
- Decimal
- Thousandths
- Hundredths
- Tenths
- Symbol
- Greater than
- Less than
- Equal
- Place value strategy
- Expanded form
- Expanded notation
MA19.5.4a
Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.
COS Examples
Example: $347.392 = 3 \times 100 + 4 \times 10 + 7 \times 1 + 3 \times (\frac{1}{10}) + 9 \times (\frac{1}{100}) + 2 \times (\frac{1}{1000})$.
Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.
COS Examples
Example: $347.392 = 3 \times 100 + 4 \times 10 + 7 \times 1 + 3 \times (\frac{1}{10}) + 9 \times (\frac{1}{100}) + 2 \times (\frac{1}{1000})$.
MA19.5.4b
Compare two decimals to thousandths based on the meaning of the digits in each place, using >, =, and < to record the results of comparisons.
Compare two decimals to thousandths based on the meaning of the digits in each place, using >, =, and < to record the results of comparisons.
MA19.5.5
Use place value understanding to round decimals to thousandths.
Use place value understanding to round decimals to thousandths.
Unpacked Content
Knowledge
Students know:
- how to use place value understanding to round multi-digit whole numbers to any place.
Skills
Students are able to:
- Round decimals using place value understanding.
Understanding
Students understand that:
- in the base ten system, the adjacent place value relationship extends to decimals and is used to round decimals.
Vocabulary
- Round
- Place value
- Tenths
- Hundredths
- Thousandths
- Decimal
- Number line
- Midpoint
MA19.5.6
Fluently multiply multi-digit whole numbers using the standard algorithm.
Fluently multiply multi-digit whole numbers using the standard algorithm.
Unpacked Content
Knowledge
Students know:
- Strategies based on place value and properties of operations for finding products of two factors including a one-digit and up to a four-digit factor and two two-digit factors.
- Decomposition of a given number into base ten units.
- How to illustrate a product of two factors using an area model.
- Connections between an area model and finding partial products when multiplying.
Skills
Students are able to:
- Use the standard algorithm to find a product.
Understanding
Students understand that:
- properties of operations and the base ten system are foundational to the computation of products using the standard algorithm.
Vocabulary
- Multiply
- Multi-digit
- Standard algorithm
- Distributive property
- Partial product
- Area model
MA19.5.7
Use strategies based on place value, properties of operations, and/or the relationship between multiplication and division to find whole-number quotients and remainders with up to four-digit dividends and two-digit divisors. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Use strategies based on place value, properties of operations, and/or the relationship between multiplication and division to find whole-number quotients and remainders with up to four-digit dividends and two-digit divisors. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Unpacked Content
Knowledge
Students know:
- Efficient strategies to find a whole number quotient when a multi-digit number (up to 4-digit dividend) is divided by a single-digit divisor.
- How to justify quotients using an illustration or the relationship between multiplication and division.
Skills
Students are able to:
- Find whole number quotients and remainders using a variety of strategies based on place value and properties of operations.
- Illustrate and explain the calculation using equations, arrays, and area models.
Understanding
Students understand that:
- Strategies for division by a one-digit divisor are extended to two-digit divisors.
- Visual models are used to illustrate division.
- Remainders may be written as a fraction or decimal and interpreted based on context of the problem situation.
Vocabulary
- Quotient
- Dividend
- Divisor
- Divide
- Multiply
- Equation
- Remainder
- Multiple
- Area model
- Decompose
- Partial quotient
- Equation
MA19.5.8
Add, subtract, multiply, and divide decimals to hundredths using strategies based on place value, properties of operations, and/or the relationships between addition/subtraction and multiplication/division; relate the strategy to a written method, and explain the reasoning used.
Add, subtract, multiply, and divide decimals to hundredths using strategies based on place value, properties of operations, and/or the relationships between addition/subtraction and multiplication/division; relate the strategy to a written method, and explain the reasoning used.
Unpacked Content
Knowledge
Students know:
- Strategies based on place value understanding, properties, and relationship between operations to find the sum, difference, product, and quotient of whole numbers.
- How to write decimal notation for fractions with denominators of 10 or 100.
- Use estimation strategies to assess reasonableness of answers.
Skills
Students are able to:
- Use concrete models, drawings, and strategies to add, subtract, multiply, and divide decimals.
- Relate strategies for operations with decimals to a written method and explain reasoning used.
- Solve real-world context problems involving decimals.
Understanding
Students understand that:
Problems involving operations with decimals
Problems involving operations with decimals
- Can be solved using a variety of strategies based on place value, properties of operations, or the relationship between the operations.
- Can be illustrated using concrete models or drawings.
Vocabulary
- Decimal
- Tenths
- Hundredths
- Place value
MA19.5.8a
Use concrete models and drawings to solve problems with decimals to hundredths.
Use concrete models and drawings to solve problems with decimals to hundredths.
MA19.5.8b
Solve problems in a real-world context with decimals to hundredths.
Solve problems in a real-world context with decimals to hundredths.
MA19.5.9
Model and solve real-word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally, and assess the reasonableness of answers.
COS Examples
Example: Recognize an incorrect result $\frac{2}{5} + \frac{1}{2} = \frac{3}{7}$ by observing that $\frac{3}{7} < \frac{1}{2}$.
Model and solve real-word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally, and assess the reasonableness of answers.
COS Examples
Example: Recognize an incorrect result $\frac{2}{5} + \frac{1}{2} = \frac{3}{7}$ by observing that $\frac{3}{7} < \frac{1}{2}$.
Unpacked Content
Knowledge
Students know:
- The meaning and magnitude of fractions expressed in units of halves, fourths, eighths, thirds, sixths, twelfths, fifths, tenths, and hundredths.
- Strategies to find sums of two or more fractions with like denominators.
- Strategies to find the difference of two fractions with like denominators.
- How to decompose a fraction greater than 1 and express as a mixed number.
Example: 7/3 = 3/3 + 3/3 + 1/3 = 2 1/3.
Skills
Students are able to:
- Solve real-word problems involving addition and subtraction of fractions with unlike denominators.
- Represent problems using fraction models or equations.
- Assess reasonableness of answers using estimation and benchmark fractions.
Understanding
Students understand that:
- solving word problems involving addition and subtraction of fractions with unlike units
- Require strategies to find equivalent fractions in a common unit, and the sum or difference will be expressed in the common unit.
- Can be assessed for reasonableness of answers using estimation strategies.
Vocabulary
- Fraction
- Benchmark fraction
- Denominator
- Fraction model
- Estimate
- Reasonableness
- Equation
- Unlike denominator
- Unlike units
MA19.5.10
Add and subtract fractions and mixed numbers with unlike denominators, using fraction equivalence to calculate a sum or difference of fractions or mixed numbers with like denominators.
Add and subtract fractions and mixed numbers with unlike denominators, using fraction equivalence to calculate a sum or difference of fractions or mixed numbers with like denominators.
Unpacked Content
Knowledge
Students know:
- Strategies to determine if two given fractions are equivalent.
- How to use a visual model to illustrate fraction equivalency.
- Contextual situations for addition and subtraction.
Skills
Students are able to:
- Use fraction equivalence to add and subtract fractions and mixed numbers with unlike denominators.
Understanding
Students understand that:
Addition and subtraction of fractions and mixed numbers with unlike units,
Addition and subtraction of fractions and mixed numbers with unlike units,
- Require strategies to find equivalent fractions in a common unit, and the sum or difference will be expressed in the common unit.
- Can be assessed for reasonableness of answers using estimation strategies.
Vocabulary
- Fraction
- Denominator
- Numerator
- Visual Model
- Sum
- Difference
- Equivalence
- Unlike denominators
- Unlike units
MA19.5.11
Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.
Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.
Unpacked Content
Knowledge
Students know:
- Contextual situations for division.
- Strategies to equipartition.
Skills
Students are able to:
- Solve word problems involving division of whole numbers leading to quotients with fractions.
- Use fraction models, drawings, equations to represent word problems.
- Model and interpret a fraction as division.
Understanding
Students understand that:
- a ÷ b is a division expression and can be written as a/b showing division of the numerator by the denominator (including cases where the value of a
Vocabulary
- Fraction
- Numerator
- Denominator
- Division
- Remainder
- Dividend
- Divisor
MA19.5.11a
Model and interpret a fraction as division of the numerator by the denominator ($\frac{a}{b} = a \div b$).
Model and interpret a fraction as division of the numerator by the denominator ($\frac{a}{b} = a \div b$).
MA19.5.11b
Use visual fraction models, drawings, or equations to represent word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.
Use visual fraction models, drawings, or equations to represent word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.
MA19.5.12
Apply and extend previous understandings of multiplication to find the product of a fraction times a whole number or a fraction times a fraction.
Apply and extend previous understandings of multiplication to find the product of a fraction times a whole number or a fraction times a fraction.
Unpacked Content
Knowledge
Students know:
- How to write an equation involving repeated addition with fractions as a multiplication equation of a whole number times the fraction.
Example: 2/9 + 2/9 + 2/9 + 2/9 = 4 x 2/9 = 8/9. - The relationship of partial products to an area model when multiplying by two whole numbers.
- Area of a rectangle is determined by multiplying side lengths and is found in square units.
Skills
Students are able to:
- Use previous understandings of multiplication to
- Find products of a fraction times a whole number and products of a fraction times a fraction.
- Use area models, linear models or set models to represent products.
- Create a story context to represent equations (a/b) × q and (a/b) × (c/d) to interpret products.
- Find area of rectangles with fractional side lengths and represent products as rectangular areas.
- Find the area of a rectangle by tiling the area of a rectangle with unit squares.
Understanding
Students understand that:
- Any whole number can be written as a fraction.
- The general rule for multiplication involving fractions can be justified through visual models.
- A variety of contextual situations can be represented by multiplication involving fractions.
- Tiling with unit squares can be used to find the area of a rectangle with fractional side lengths.
Vocabulary
- Fraction
- Fraction model
- Whole number
- Area
- Area model
- Linear model
- Set model
- Tiling
- Unit squares
- Equation
MA19.5.12a
Use a visual fraction model (area model, set model, or linear model) to show $(\frac{a}{b}) \times q$ and create a story context for this equation to interpret the product as a parts of a partition of q into b equal parts.
Use a visual fraction model (area model, set model, or linear model) to show $(\frac{a}{b}) \times q$ and create a story context for this equation to interpret the product as a parts of a partition of q into b equal parts.
MA19.5.12b
Use a visual fraction model (area model, set model, or linear model) to show $(\frac{a}{b}) \times (\frac{c}{d})$ and create a story context for this equation to interpret the product.
Use a visual fraction model (area model, set model, or linear model) to show $(\frac{a}{b}) \times (\frac{c}{d})$ and create a story context for this equation to interpret the product.
MA19.5.12c
Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
MA19.5.12d
Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths to show that the area is the same as would be found by multiplying the side lengths.
Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths to show that the area is the same as would be found by multiplying the side lengths.
MA19.5.13
Interpret multiplication as scaling (resizing).
Interpret multiplication as scaling (resizing).
Unpacked Content
Knowledge
Students know:
- How to interpret multiplicative comparisons.
- Strategies to compare products with whole numbers using reasoning and justification.
Example: Which is greater? 5 x 2 x 13 or 13 x 9? 10 x 13 is greater than 9 x 13 because both expressions contain a factor of 13, but the scale factor of 10 will result in a greater product than a scale factor of 9. - Fraction meaning and magnitude of fractions less than and greater than 1.
Skills
Students are able to:
- Interpret multiplication as scaling.
- Use reasoning to compare products of multiplication expressions.
- Reason and explain when multiplying a given number by a fraction why the product will be greater than or less than the original number.
Understanding
Students understand that:
- a product reflects the size of its factors.
Vocabulary
- Resizing
- Scaling
- Product
- Factor
MA19.5.13a
Compare the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
COS Examples
Example: Use reasoning to determine which expression is greater? 225 or $\frac{3}{4} \times 225$ ; $\frac{11}{50}$ or $\frac{3}{2} \times \frac{11}{50}$
Compare the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
COS Examples
Example: Use reasoning to determine which expression is greater? 225 or $\frac{3}{4} \times 225$ ; $\frac{11}{50}$ or $\frac{3}{2} \times \frac{11}{50}$
MA19.5.13b
Explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number and relate the principle of fraction equivalence.
Explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number and relate the principle of fraction equivalence.
MA19.5.13c
Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number and relate the principle of fraction equivalence.
Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number and relate the principle of fraction equivalence.
MA19.5.14
Model and solve real-world problems involving multiplication of fractions and mixed numbers using visual fraction models, drawings, or equations to represent the problem.
Model and solve real-world problems involving multiplication of fractions and mixed numbers using visual fraction models, drawings, or equations to represent the problem.
Unpacked Content
Knowledge
Students know:
- Contextual situations for multiplication.
- How to use an area model to illustrate the product of two whole numbers and its relationship to partial products and extend this knowledge to illustrate products involving fractions and mixed numbers.
Skills
Students are able to:
- Solve real-word problems involving multiplication of fractions and mixed numbers.
- Write equations to represent the word situation.
- Use visual fraction models to represent the problem.
Understanding
Students understand that:
- A variety of strategies are used to model and solve problems that provide a context for multiplying fractions and mixed numbers.
- Solutions are interpreted based on the meaning of the quantities and the context of the problem situation.
Vocabulary
- Fraction
- Models
- Mixed number
- Multiplication
