Foundational Skills in Mathematics 3-5 (#5012015) 


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Course Standards


Name Description
MA.3.AR.1.1: Apply the distributive property to multiply a one-digit number and two-digit number. Apply properties of multiplication to find a product of one-digit whole numbers.
Clarifications:
Clarification 1: Within this benchmark, the expectation is to apply the associative and commutative properties of multiplication, the distributive property and name the properties. Refer to K-12 Glossary (Appendix C).

Clarification 2: Within the benchmark, the expectation is to utilize parentheses. 

Clarification 3: Multiplication for products of three or more numbers is limited to factors within 12. Refer to Properties of Operations, Equality and Inequality (Appendix D).


Examples:
The product 4×72 can be found by rewriting the expression as 4×(70+2) and then using the distributive property to obtain (4×70)+(4×2) which is equivalent to 288.
MA.3.AR.1.2: Solve one- and two-step real-world problems involving any of four operations with whole numbers.
Clarifications:
Clarification 1: Instruction includes understanding the context of the problem, as well as the quantities within the problem.

Clarification 2: Multiplication is limited to factors within 12 and related division facts. Refer to Situations Involving Operations with Numbers (Appendix A).


Examples:
A group of students are playing soccer during lunch. How many students are needed to form four teams with eleven players each and to have two referees?
MA.3.AR.2.1: Restate a division problem as a missing factor problem using the relationship between multiplication and division.
Clarifications:
Clarification 1: Multiplication is limited to factors within 12 and related division facts.

Clarification 2: Within this benchmark, the symbolic representation of the missing factor uses any symbol or a letter.


Examples:
The equation 56÷7=? can be restated as 7×?=56 to determine the quotient is 8.
MA.3.AR.2.2: Determine and explain whether an equation involving multiplication or division is true or false.
Clarifications:
Clarification 1: Instruction extends the understanding of the meaning of the equal sign to multiplication and division.

Clarification 2: Problem types are limited to an equation with three or four terms. The product or quotient can be on either side of the equal sign. 

Clarification 3: Multiplication is limited to factors within 12 and related division facts.


Examples:
Given the equation 27÷3=3×3 , it can be determined to be a true equation by dividing the numbers on the left side of the equal sign and multiplying the numbers on the right of the equal sign to see that both sides are equivalent to 9.
MA.3.AR.2.3: Determine the unknown whole number in a multiplication or division equation, relating three whole numbers, with the unknown in any position.
Clarifications:
Clarification 1: Instruction extends the development of algebraic thinking skills where the symbolic representation of the unknown uses any symbol or a letter. 
Clarification 2: Problems include the unknown on either side of the equal sign. 
Clarification 3: Multiplication is limited to factors within 12 and related division facts. Refer to Situations Involving Operations with Numbers (Appendix A).
MA.3.AR.3.1: Determine and explain whether a whole number from 1 to 1,000 is even or odd.
Clarifications:
Clarification 1: Instruction includes determining and explaining using place value and recognizing patterns.
MA.3.AR.3.2: Determine whether a whole number from 1 to 144 is a multiple of a given one-digit number.
Clarifications:
Clarification 1: Instruction includes determining if a number is a multiple of a given number by using multiplication or division.
MA.3.AR.3.3: Identify, create and extend numerical patterns.
Clarifications:
Clarification 1: The expectation is to use ordinal numbers (1st, 2nd, 3rd, …) to describe the position of a number within a sequence.

Clarification 2: Problem types include patterns involving addition, subtraction, multiplication or division of whole numbers.


Examples:
Bailey collects 6 baseball cards every day. This generates the pattern 6,12,18,… How many baseball cards will Bailey have at the end of the sixth day?
MA.3.DP.1.1: Collect and represent numerical and categorical data with whole-number values using tables, scaled pictographs, scaled bar graphs or line plots. Use appropriate titles, labels and units.
Clarifications:
Clarification 1: Within this benchmark, the expectation is to complete a representation or construct a representation from a data set.

 Clarification 2: Instruction includes the connection between multiplication and the number of data points represented by a bar in scaled bar graph or a scaled column in a pictograph.

Clarification 3: Data displays are represented both horizontally and vertically.

MA.3.DP.1.2: Interpret data with whole-number values represented with tables, scaled pictographs, circle graphs, scaled bar graphs or line plots by solving one- and two-step problems.
Clarifications:
Clarification 1: Problems include the use of data in informal comparisons between two data sets in the same units.

 Clarification 2: Data displays can be represented both horizontally and vertically.

Clarification 3: Circle graphs are limited to showing the total values in each category.

MA.3.FR.1.1: Represent and interpret unit fractions in the form 1/n as the quantity formed by one part when a whole is partitioned into n equal parts.
Clarifications:
Clarification 1: This benchmark emphasizes conceptual understanding through the use of manipulatives or visual models. 
Clarification 2: Instruction focuses on representing a unit fraction as part of a whole, part of a set, a point on a number line, a visual model or in fractional notation.

Clarification 3: Denominators are limited to 2, 3, 4, 5, 6, 8, 10 and 12.


Examples:
begin mathsize 11px style 1 fourth end style can be represented as begin mathsize 12px style 1 fourth end style of a pie (parts of a shape), as 1 out of 4 trees (parts of a set) or as begin mathsize 12px style 1 fourth end style on the number line.
MA.3.FR.1.2: Represent and interpret fractions, including fractions greater than one, in the form of mn as the result of adding the unit fraction 1n to itself times.
Clarifications:
Clarification 1: Instruction emphasizes conceptual understanding through the use of manipulatives or visual models, including circle graphs, to represent fractions.

Clarification 2: Denominators are limited to 2, 3, 4, 5, 6, 8, 10 and 12.


Examples:
begin mathsize 12px style 9 over 8 end style can be represented as begin mathsize 12px style 1 over 8 plus 1 over 8 plus 1 over 8 plus 1 over 8 plus 1 over 8 plus 1 over 8 plus 1 over 8 plus 1 over 8 plus 1 over 8 end style.
MA.3.FR.1.3: Read and write fractions, including fractions greater than one, using standard form, numeral-word form and word form.
Clarifications:
Clarification 1: Instruction focuses on making connections to reading and writing numbers to develop the understanding that fractions are numbers and to support algebraic thinking in later grades.

Clarification 2: Denominators are limited to 2, 3, 4, 5, 6, 8, 10 and 12.


Examples:
The fraction begin mathsize 12px style 4 over 3 end style written in word form is four-thirds and in numeral-word form is 4 thirds.
MA.3.FR.2.1: Plot, order and compare fractional numbers with the same numerator or the same denominator.
Clarifications:
Clarification 1: Instruction includes making connections between using a ruler and plotting and ordering fractions on a number line. 
Clarification 2: When comparing fractions, instruction includes an appropriately scaled number line and using reasoning about their size.

Clarification 3: Fractions include fractions greater than one, including mixed numbers, with denominators limited to 2, 3, 4, 5, 6, 8, 10 and 12.


Examples:
The fraction begin mathsize 12px style 3 over 2 end style is to the right of the fraction begin mathsize 12px style 3 over 3 end style on a number line so begin mathsize 12px style 3 over 2 end style is greater than begin mathsize 12px style 3 over 3 end style.
MA.3.FR.2.2: Identify equivalent fractions and explain why they are equivalent.
Clarifications:
Clarification 1: Instruction includes identifying equivalent fractions and explaining why they are equivalent using manipulatives, drawings, and number lines.

Clarification 2: Within this benchmark, the expectation is not to generate equivalent fractions. 

Clarification 3: Fractions are limited to fractions less than or equal to one with denominators of 2, 3, 4, 5, 6, 8, 10 and 12. Number lines must be given and scaled appropriately.


Examples:
Example: The fractions begin mathsize 12px style 1 over 1 end style and begin mathsize 12px style 3 over 3 end style can be identified as equivalent using number lines.

Example: The fractions begin mathsize 12px style 2 over 4 end style and begin mathsize 12px style 2 over 6 end style can be identified as not equivalent using a visual model.

MA.3.GR.1.1: Describe and draw points, lines, line segments, rays, intersecting lines, perpendicular lines and parallel lines. Identify these in two-dimensional figures.
Clarifications:
Clarification 1: Instruction includes mathematical and real-world context for identifying points, lines, line segments, rays, intersecting lines, perpendicular lines and parallel lines.

Clarification 2: When working with perpendicular lines, right angles can be called square angles or square corners.

MA.3.GR.1.2: Identify and draw quadrilaterals based on their defining attributes. Quadrilaterals include parallelograms, rhombi, rectangles, squares and trapezoids.
Clarifications:
Clarification 1: Instruction includes a variety of quadrilaterals and a variety of non-examples that lack one or more defining attributes when identifying quadrilaterals.

Clarification 2: Quadrilaterals will be filled, outlined or both when identifying.

Clarification 3: Drawing representations must be reasonably accurate.

MA.3.GR.1.3: Draw line(s) of symmetry in a two-dimensional figure and identify line-symmetric two-dimensional figures.
Clarifications:
Clarification 1: Instruction develops the understanding that there could be no line of symmetry, exactly one line of symmetry or more than one line of symmetry.

Clarification 2: Instruction includes folding paper along a line of symmetry so that both halves match exactly to confirm line-symmetric figures.

MA.3.GR.2.1: Explore area as an attribute of a two-dimensional figure by covering the figure with unit squares without gaps or overlaps. Find areas of rectangles by counting unit squares.
Clarifications:
Clarification 1: Instruction emphasizes the conceptual understanding that area is an attribute that can be measured for a two-dimensional figure. The measurement unit for area is the area of a unit square, which is a square with side length of 1 unit.

Clarification 2: Two-dimensional figures cannot exceed 12 units by 12 units and responses include the appropriate units in word form (e.g., square centimeter or sq.cm.).

MA.3.GR.2.2: Find the area of a rectangle with whole-number side lengths using a visual model and a multiplication formula.
Clarifications:
Clarification 1: Instruction includes covering the figure with unit squares, a rectangular array or applying a formula.

Clarification 2: Two-dimensional figures cannot exceed 12 units by 12 units and responses include the appropriate units in word form.

MA.3.GR.2.3: Solve mathematical and real-world problems involving the perimeter and area of rectangles with whole-number side lengths using a visual model and a formula.
Clarifications:
Clarification 1: Within this benchmark, the expectation is not to find unknown side lengths.

Clarification 2: Two-dimensional figures cannot exceed 12 units by 12 units and responses include the appropriate units in word form.

MA.3.GR.2.4: Solve mathematical and real-world problems involving the perimeter and area of composite figures composed of non-overlapping rectangles with whole-number side lengths.
Clarifications:
Clarification 1: Composite figures must be composed of non-overlapping rectangles.

Clarification 2: Each rectangle within the composite figure cannot exceed 12 units by 12 units and responses include the appropriate units in word form.


Examples:
A pool is comprised of two non-overlapping rectangles in the shape of an “L”. The area for a cover of the pool can be found by adding the areas of the two non-overlapping rectangles.
MA.3.M.1.1: Select and use appropriate tools to measure the length of an object, the volume of liquid within a beaker and temperature.
Clarifications:
Clarification 1: Instruction focuses on identifying measurement on a linear scale, making the connection to the number line.

Clarification 2: When measuring the length, limited to the nearest centimeter and half or quarter inch.

Clarification 3: When measuring the temperature, limited to the nearest degree.

Clarification 4: When measuring the volume of liquid, limited to nearest milliliter and half or quarter cup.

MA.3.M.1.2: Solve real-world problems involving any of the four operations with whole-number lengths, masses, weights, temperatures or liquid volumes.
Clarifications:
Clarification 1: Within this benchmark, it is the expectation that responses include appropriate units.

Clarification 2: Problem types are not expected to include measurement conversions.

Clarification 3: Instruction includes the comparison of attributes measured in the same units.

Clarification 4: Units are limited to yards, feet, inches; meters, centimeters; pounds, ounces; kilograms, grams; degrees Fahrenheit, degrees Celsius; gallons, quarts, pints, cups; and liters, milliliters.


Examples:
Ms. Johnson’s class is having a party. Eight students each brought in a 2-liter bottle of soda for the party. How many liters of soda did the class have for the party?
MA.3.M.2.1: Using analog and digital clocks tell and write time to the nearest minute using a.m. and p.m. appropriately.
Clarifications:
Clarification 1: Within this benchmark, the expectation is not to understand military time.
MA.3.M.2.2: Solve one- and two-step real-world problems involving elapsed time.
Clarifications:
Clarification 1: Within this benchmark, the expectation is not to include crossing between a.m. and p.m.

Examples:
A bus picks up Kimberly at 6:45 a.m. and arrives at school at 8:15 a.m. How long was her bus ride?
MA.3.NSO.1.1: Read and write numbers from 0 to 10,000 using standard form, expanded form and word form.
Examples:
The number two thousand five hundred thirty written in standard form is 2,530 and in expanded form is 2,000+500+30.
MA.3.NSO.1.2: Compose and decompose four-digit numbers in multiple ways using thousands, hundreds, tens and ones. Demonstrate each composition or decomposition using objects, drawings and expressions or equations.
Examples:
The number 5,783 can be expressed as 5 thousands + 7 hundreds + 8 tens + 3 ones or as 56 hundreds + 183 ones.
MA.3.NSO.1.3: Plot, order and compare whole numbers up to 10,000.
Clarifications:
Clarification 1: When comparing numbers, instruction includes using an appropriately scaled number line and using place values of the thousands, hundreds, tens and ones digits.

Clarification 2: Number lines, scaled by 50s, 100s or 1,000s, must be provided and can be a representation of any range of numbers.

Clarification 3: Within this benchmark, the expectation is to use symbols (<, > or =).


Examples:
The numbers 3,475; 4,743 and 4,753 can be arranged in ascending order as 3,475; 4,743 and 4,753.
MA.3.NSO.1.4: Round whole numbers from 0 to 1,000 to the nearest 10 or 100.
Examples:
Example: The number 775 is rounded to 780 when rounded to the nearest 10.

Example: The number 745 is rounded to 700 when rounded to the nearest 100.

MA.3.NSO.2.1: Add and subtract multi-digit whole numbers including using a standard algorithm with procedural fluency.
MA.3.NSO.2.2: Explore multiplication of two whole numbers with products from 0 to 144, and related division facts.
Clarifications:
Clarification 1: Instruction includes equal groups, arrays, area models and equations.

Clarification 2: Within the benchmark, it is the expectation that one problem can be represented in multiple ways and understanding how the different representations are related to each other.

Clarification 3: Factors and divisors are limited to up to 12.

MA.3.NSO.2.3: Multiply a one-digit whole number by a multiple of 10, up to 90, or a multiple of 100, up to 900, with procedural reliability.
Clarifications:
Clarification 1: When multiplying one-digit numbers by multiples of 10 or 100, instruction focuses on methods that are based on place value.

Examples:
Example: The product of 6 and 70 is 420.

Example: The product of 6 and 300 is 1,800.

MA.3.NSO.2.4: Multiply two whole numbers from 0 to 12 and divide using related facts with procedural reliability.
Clarifications:
Clarification 1: Instruction focuses on helping a student choose a method they can use reliably.

Examples:
Example: The product of 5 and 6 is 30.

Example: The quotient of 27 and 9 is 3.

MA.4.AR.1.1: Solve real-world problems involving multiplication and division of whole numbers including problems in which remainders must be interpreted within the context.
Clarifications:
Clarification 1: Problems involving multiplication include multiplicative comparisons. Refer to Situations Involving Operations with Numbers (Appendix A).

Clarification 2: Depending on the context, the solution of a division problem with a remainder may be the whole number part of the quotient, the whole number part of the quotient with the remainder, the whole number part of the quotient plus 1, or the remainder.

Clarification 3: Multiplication is limited to products of up to 3 digits by 2 digits. Division is limited to up to 4 digits divided by 1 digit.


Examples:
A group of 243 students is taking a field trip and traveling in vans. If each van can hold 8 students, then the group would need 31 vans for their field trip because 243 divided by 8 equals 30 with a remainder of 3.
MA.4.AR.1.2: Solve real-world problems involving addition and subtraction of fractions with like denominators, including mixed numbers and fractions greater than one.
Clarifications:
Clarification 1: Problems include creating real-world situations based on an equation or representing a real-world problem with a visual model or equation.

Clarification 2: Fractions within problems must reference the same whole.

Clarification 3: Within this benchmark, the expectation is not to simplify or use lowest terms.

Clarification 4: Denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, 16 and 100.


Examples:
Example: Megan is making pies and uses the equation begin mathsize 12px style 1 3 over 4 plus 3 1 fourth equals x end style when baking. Describe a situation that can represent this equation.

Example: Clay is running a 10K race. So far, he has run begin mathsize 12px style 6 1 fifth end style kilometers. How many kilometers does he have remaining?

MA.4.AR.1.3: Solve real-world problems involving multiplication of a fraction by a whole number or a whole number by a fraction.
Clarifications:
Clarification 1: Problems include creating real-world situations based on an equation or representing a real-world problem with a visual model or equation.

Clarification 2: Fractions within problems must reference the same whole.

Clarification 3: Within this benchmark, the expectation is not to simplify or use lowest terms.

Clarification 4: Fractions limited to fractions less than one with denominators of 2, 3, 4, 5, 6, 8, 10, 12, 16 and 100.


Examples:
Ken is filling his garden containers with a cup that holds begin mathsize 12px style 2 over 5 end style pounds of soil. If he uses 8 cups to fill his garden containers, how many pounds of soil did Ken use?
MA.4.AR.2.1: Determine and explain whether an equation involving any of the four operations with whole numbers is true or false.
Clarifications:
Clarification 1: Multiplication is limited to whole number factors within 12 and related division facts.

Examples:
The equation 32÷8=32-8-8-8-8 can be determined to be false because the expression on the left side of the equal sign is not equivalent to the expression on the right side of the equal sign.
MA.4.AR.2.2: Given a mathematical or real-world context, write an equation involving multiplication or division to determine the unknown whole number with the unknown in any position.
Clarifications:
Clarification 1: Instruction extends the development of algebraic thinking skills where the symbolic representation of the unknown uses a letter.

Clarification 2: Problems include the unknown on either side of the equal sign.

Clarification 3: Multiplication is limited to factors within 12 and related division facts.


Examples:
The equation 96=8×t can be used to determine the cost of each movie ticket at the movie theatre if a total of $96 was spent on 8 equally priced tickets. Then each ticket costs $12.
MA.4.AR.3.1: Determine factor pairs for a whole number from 0 to 144. Determine whether a whole number from 0 to 144 is prime, composite or neither.
Clarifications:
Clarification 1: Instruction includes the connection to the relationship between multiplication and division and patterns with divisibility rules.

Clarification 2: The numbers 0 and 1 are neither prime nor composite.

MA.4.AR.3.2: Generate, describe and extend a numerical pattern that follows a given rule.
Clarifications:
Clarification 1: Instruction includes patterns within a mathematical or real-world context.

Examples:
Generate a pattern of four numbers that follows the rule of adding 14 starting at 5.
MA.4.DP.1.1: Collect and represent numerical data, including fractional values, using tables, stem-and-leaf plots or line plots.
Clarifications:
Clarification 1: Denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 16 and 100.

Examples:
A softball team is measuring their hat size. Each player measures the distance around their head to the nearest half inch. The data is collected and represented on a line plot.
MA.4.DP.1.2: Determine the mode, median or range to interpret numerical data including fractional values, represented with tables, stem-and-leaf plots or line plots.
Clarifications:
Clarification 1: Instruction includes interpreting data within a real-world context.

Clarification 2: Instruction includes recognizing that data sets can have one mode, no mode or more than one mode.

Clarification 3: Within this benchmark, data sets are limited to an odd number when calculating the median.

Clarification 4: Denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 16 and 100.


Examples:
Given the data of the softball team’s hat size represented on a line plot, determine the most common size and the difference between the largest and the smallest sizes.
MA.4.DP.1.3: Solve real-world problems involving numerical data.
Clarifications:
Clarification 1: Instruction includes using any of the four operations to solve problems.

Clarification 2: Data involving fractions with like denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 16 and 100. Fractions can be greater than one.

Clarification 3: Data involving decimals are limited to hundredths.


Examples:
Given the data of the softball team’s hat size represented on a line plot, determine the fraction of the team that has a head size smaller than 20 inches.
MA.4.FR.1.1: Model and express a fraction, including mixed numbers and fractions greater than one, with the denominator 10 as an equivalent fraction with the denominator 100.
Clarifications:
Clarification 1: Instruction emphasizes conceptual understanding through the use of manipulatives, visual models, number lines or equations.
MA.4.FR.1.2: Use decimal notation to represent fractions with denominators of 10 or 100, including mixed numbers and fractions greater than 1, and use fractional notation with denominators of 10 or 100 to represent decimals.
Clarifications:
Clarification 1: Instruction emphasizes conceptual understanding through the use of manipulatives visual models, number lines or equations.

Clarification 2: Instruction includes the understanding that a decimal and fraction that are equivalent represent the same point on the number line and that fractions with denominators of 10 or powers of 10 may be called decimal fractions.

MA.4.FR.1.3: Identify and generate equivalent fractions, including fractions greater than one. Describe how the numerator and denominator are affected when the equivalent fraction is created.
Clarifications:
Clarification 1: Instruction includes the use of manipulatives, visual models, number lines or equations.

Clarification 2: Instruction includes recognizing how the numerator and denominator are affected when equivalent fractions are generated.

MA.4.FR.1.4: Plot, order and compare fractions, including mixed numbers and fractions greater than one, with different numerators and different denominators.
Clarifications:
Clarification 1: When comparing fractions, instruction includes using an appropriately scaled number line and using reasoning about their size.

Clarification 2: Instruction includes using benchmark quantities, such as 0, begin mathsize 12px style 1 fourth end style, begin mathsize 12px style 1 half end style, begin mathsize 12px style 3 over 4 end style and 1, to compare fractions.

Clarification 3: Denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 16 and 100.

Clarification 4: Within this benchmark, the expectation is to use symbols (<, > or =).


Examples:
begin mathsize 12px style 1 2 over 3 greater than 1 1 fourth end style because begin mathsize 12px style 2 over 3 end style is greater than begin mathsize 12px style 1 half end style and begin mathsize 12px style 1 half end style is greater than begin mathsize 12px style 1 fourth end style.
MA.4.FR.2.1: Decompose a fraction, including mixed numbers and fractions greater than one, into a sum of fractions with the same denominator in multiple ways. Demonstrate each decomposition with objects, drawings and equations.
Clarifications:
Clarification 1: Denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 16 and 100.

Examples:
begin mathsize 12px style 9 over 8 end style can be decomposed as begin mathsize 12px style 8 over 8 plus 1 over 8 end style or as begin mathsize 12px style 3 over 8 plus 3 over 8 plus 3 over 8 end style.
MA.4.FR.2.2: Add and subtract fractions with like denominators, including mixed numbers and fractions greater than one, with procedural reliability.
Clarifications:
Clarification 1: Instruction includes the use of word form, manipulatives, drawings, the properties of operations or number lines.

Clarification 2: Within this benchmark, the expectation is not to simplify or use lowest terms.

Clarification 3: Denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 16 and 100.


Examples:
The difference begin mathsize 12px style 9 over 5 minus 4 over 5 end stylecan be expressed as 9 fifths minus 4 fifths which is 5 fifths, or one.
MA.4.FR.2.3: Explore the addition of a fraction with denominator of 10 to a fraction with denominator of 100 using equivalent fractions.
Clarifications:
Clarification 1: Instruction includes the use of visual models.

Clarification 2: Within this benchmark, the expectation is not to simplify or use lowest terms.


Examples:
begin mathsize 12px style 9 over 100 plus 3 over 10 end style is equivalent to begin mathsize 12px style 9 over 100 plus 30 over 100 end style which is equivalent to begin mathsize 12px style 39 over 100 end style.
MA.4.FR.2.4: Extend previous understanding of multiplication to explore the multiplication of a fraction by a whole number or a whole number by a fraction.
Clarifications:
Clarification 1: Instruction includes the use of visual models or number lines and the connection to the commutative property of multiplication. Refer to Properties of Operation, Equality and Inequality (Appendix D).

Clarification 2: Within this benchmark, the expectation is not to simplify or use lowest terms.

Clarification 3: Fractions multiplied by a whole number are limited to less than 1. All denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 16, 100.


Examples:
Example: Shanice thinks about finding the product begin mathsize 12px style 1 fourth cross times 8 end style by imagining having 8 pizzas that she wants to split equally with three of her friends. She and each of her friends will get 2 pizzas since begin mathsize 12px style 1 fourth cross times 8 equals 2 end style.

Example: Lacey thinks about finding the product begin mathsize 12px style 8 cross times 1 fourth end style by imagining having 8 pizza boxes each with one-quarter slice of a pizza left. If she put them all together, she would have a total of 2 whole pizzas since begin mathsize 12px style 8 cross times 1 fourth equals 8 over 4 end style which is equivalent to 2.

MA.4.GR.1.1: Informally explore angles as an attribute of two-dimensional figures. Identify and classify angles as acute, right, obtuse, straight or reflex.
Clarifications:
Clarification 1: Instruction includes classifying angles using benchmark angles of 90° and 180° in two-dimensional figures.

Clarification 2: When identifying angles, the expectation includes two-dimensional figures and real-world pictures.

MA.4.GR.1.2: Estimate angle measures. Using a protractor, measure angles in whole-number degrees and draw angles of specified measure in whole-number degrees. Demonstrate that angle measure is additive.
Clarifications:

Clarification 1: Instruction includes measuring given angles and drawing angles using protractors.
Clarification 2: Instruction includes estimating angle measures using benchmark angles (30°, 45°, 60°, 90° and 180°).
Clarification 3: Instruction focuses on the understanding that angles can be decomposed into non-overlapping angles whose measures sum to the measure of the original angle.

MA.4.GR.1.3: Solve real-world and mathematical problems involving unknown whole-number angle measures. Write an equation to represent the unknown.
Clarifications:
Clarification 1: Instruction includes the connection to angle measure as being additive.

Examples:
A 60° angle is decomposed into two angles, one of which is 25°. What is the measure of the other angle?
MA.4.GR.2.1: Solve perimeter and area mathematical and real-world problems, including problems with unknown sides, for rectangles with whole-number side lengths.
Clarifications:
Clarification 1: Instruction extends the development of algebraic thinking where the symbolic representation of the unknown uses a letter.

Clarification 2: Problems involving multiplication are limited to products of up to 3 digits by 2 digits. Problems involving division are limited to up to 4 digits divided by 1 digit.

Clarification 3: Responses include the appropriate units in word form.

MA.4.GR.2.2: Solve problems involving rectangles with the same perimeter and different areas or with the same area and different perimeters.
Clarifications:
Clarification 1: Instruction focuses on the conceptual understanding of the relationship between perimeter and area.

Clarification 2: Within this benchmark, rectangles are limited to having whole-number side lengths.

Clarification 3: Problems involving multiplication are limited to products of up to 3 digits by 2 digits. Problems involving division are limited to up to 4 digits divided by 1 digit.

Clarification 4: Responses include the appropriate units in word form.


Examples:
Possible dimensions of a rectangle with an area of 24 square feet include 6 feet by 4 feet or 8 feet by 3 feet. This can be found by cutting a rectangle into unit squares and rearranging them.
MA.4.M.1.1: Select and use appropriate tools to measure attributes of objects.
Clarifications:
Clarification 1: Attributes include length, volume, weight, mass and temperature.

Clarification 2: Instruction includes digital measurements and scales that are not linear in appearance.

Clarification 3: When recording measurements, use fractions and decimals where appropriate.

MA.4.M.1.2: Convert within a single system of measurement using the units: yards, feet, inches; kilometers, meters, centimeters, millimeters; pounds, ounces; kilograms, grams; gallons, quarts, pints, cups; liter, milliliter; and hours, minutes, seconds.
Clarifications:
Clarification 1: Instruction includes the understanding of how to convert from smaller to larger units or from larger to smaller units.

Clarification 2: Within the benchmark, the expectation is not to convert from grams to kilograms, meters to kilometers or milliliters to liters.

Clarification 3: Problems involving fractions are limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12, 16 and 100.


Examples:
Example: If a ribbon is 11 yards 2 feet in length, how long is the ribbon in feet?

Example: A gallon contains 16 cups. How many cups are in begin mathsize 12px style 3 1 half end style gallons?

MA.4.M.2.1: Solve two-step real-world problems involving distances and intervals of time using any combination of the four operations.
Clarifications:
Clarification 1: Problems involving fractions will include addition and subtraction with like denominators and multiplication of a fraction by a whole number or a whole number by a fraction. 
Clarification 2: Problems involving fractions are limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12, 16 and 100.
Clarification 3: Within the benchmark, the expectation is not to use decimals.
MA.4.M.2.2: Solve one- and two-step addition and subtraction real-world problems involving money using decimal notation.
Examples:
Example: An item costs $1.84. If you give the cashier $2.00, how much change should you receive? What coins could be used to give the change?

Example: At the grocery store you spend $14.56. If you do not want any pennies in change, how much money could you give the cashier?

MA.4.NSO.1.1: Express how the value of a digit in a multi-digit whole number changes if the digit moves one place to the left or right.
MA.4.NSO.1.2: Read and write multi-digit whole numbers from 0 to 1,000,000 using standard form, expanded form and word form.
Examples:
The number two hundred seventy-five thousand eight hundred two written in standard form is 275,802 and in expanded form is 200,000+70,000+5,000+800+2 or (2×100,000)+(7×10,000)+(5×1,000)+(8×100)+(2×1).
MA.4.NSO.1.3: Plot, order and compare multi-digit whole numbers up to 1,000,000.
Clarifications:
Clarification 1: When comparing numbers, instruction includes using an appropriately scaled number line and using place values of the hundred thousands, ten thousands, thousands, hundreds, tens and ones digits.

Clarification 2: Scaled number lines must be provided and can be a representation of any range of numbers.

 Clarification 3: Within this benchmark, the expectation is to use symbols (<, > or =).


Examples:
The numbers 75,421; 74,241 and 74,521 can be arranged in ascending order as 74,241; 74,521 and 75,421.
MA.4.NSO.1.4: Round whole numbers from 0 to 10,000 to the nearest 10, 100 or 1,000.
Examples:
Example: The number 6,325 is rounded to 6,300 when rounded to the nearest 100.

Example: The number 2,550 is rounded to 3,000 when rounded to the nearest 1,000.

MA.4.NSO.1.5: Plot, order and compare decimals up to the hundredths.
Clarifications:
Clarification 1: When comparing numbers, instruction includes using an appropriately scaled number line and using place values of the ones, tenths and hundredths digits.

Clarification 2: Within the benchmark, the expectation is to explain the reasoning for the comparison and use symbols (<, > or =). 

Clarification 3: Scaled number lines must be provided and can be a representation of any range of numbers.


Examples:
The numbers 3.2; 3.24 and 3.12 can be arranged in ascending order as 3.12; 3.2 and 3.24.
MA.4.NSO.2.1: Recall multiplication facts with factors up to 12 and related division facts with automaticity.
MA.4.NSO.2.2: Multiply two whole numbers, up to three digits by up to two digits, with procedural reliability.
Clarifications:
Clarification 1: Instruction focuses on helping a student choose a method they can use reliably.

Clarification 2: Instruction includes the use of models or equations based on place value and the distributive property.

MA.4.NSO.2.3: Multiply two whole numbers, each up to two digits, including using a standard algorithm with procedural fluency.
MA.4.NSO.2.4: Divide a whole number up to four digits by a one-digit whole number with procedural reliability. Represent remainders as fractional parts of the divisor.
Clarifications:
Clarification 1: Instruction focuses on helping a student choose a method they can use reliably.

Clarification 2: Instruction includes the use of models based on place value, properties of operations or the relationship between multiplication and division.

MA.4.NSO.2.5: Explore the multiplication and division of multi-digit whole numbers using estimation, rounding and place value.
Clarifications:
Clarification 1: Instruction focuses on previous understanding of multiplication with multiples of 10 and 100, and seeing division as a missing factor problem.

Clarification 2: Estimating quotients builds the foundation for division using a standard algorithm.

Clarification 3: When estimating the division of whole numbers, dividends are limited to up to four digits and divisors are limited to up to two digits.


Examples:
Example: The product of 215 and 460 can be estimated as being between 80,000 and 125,000 because it is bigger than 200×400 but smaller than 250×500.

Example: The quotient of 1,380 and 27 can be estimated as 50 because 27 is close to 30 and 1,380 is close to 1,500. 1,500 divided by 30 is the same as 150 tens divided by 3 tens which is 5 tens, or 50.

MA.4.NSO.2.6: Identify the number that is one-tenth more, one-tenth less, one-hundredth more and one-hundredth less than a given number.
Examples:
Example: One-hundredth less than 1.10 is 1.09.

Example: One-tenth more than 2.31 is 2.41.

MA.4.NSO.2.7: Explore the addition and subtraction of multi-digit numbers with decimals to the hundredths.
Clarifications:
Clarification 1: Instruction includes the connection to money and the use of manipulatives and models based on place value.
MA.5.AR.1.1: Solve multi-step real-world problems involving any combination of the four operations with whole numbers, including problems in which remainders must be interpreted within the context.
Clarifications:
Clarification 1: Depending on the context, the solution of a division problem with a remainder may be the whole number part of the quotient, the whole number part of the quotient with the remainder, the whole number part of the quotient plus 1, or the remainder.
MA.5.AR.1.2: Solve real-world problems involving the addition, subtraction or multiplication of fractions, including mixed numbers and fractions greater than 1.
Clarifications:
Clarification 1: Instruction includes the use of visual models and equations to represent the problem.

Examples:
Shanice had a sleepover and her mom is making French toast in the morning. If her mom had begin mathsize 12px style 2 1 fourth end style

  loaves of bread and used begin mathsize 12px style 1 1 half end style loaves for the French toast, how much bread does she have left?

MA.5.AR.1.3: Solve real-world problems involving division of a unit fraction by a whole number and a whole number by a unit fraction.
Clarifications:
Clarification 1: Instruction includes the use of visual models and equations to represent the problem.

Examples:
Example: A property has a total of begin mathsize 12px style 1 half end style acre and needs to be divided equally among 3 sisters. Each sister will receive begin mathsize 12px style 1 over 6 end style of an acre.

Example: Kiki has 10 candy bars and plans to give begin mathsize 12px style 1 fourth end style of a candy bar to her classmates at school. How many classmates will receive a piece of a candy bar?

MA.5.AR.2.1: Translate written real-world and mathematical descriptions into numerical expressions and numerical expressions into written mathematical descriptions.
Clarifications:
Clarification 1: Expressions are limited to any combination of the arithmetic operations, including parentheses, with whole numbers, decimals and fractions. 

Clarification 2: Within this benchmark, the expectation is not to include exponents or nested grouping symbols.


Examples:
The expression 4.5 + (3×2) in word form is four and five tenths plus the quantity 3 times 2.
MA.5.AR.2.2: Evaluate multi-step numerical expressions using order of operations.
Clarifications:
Clarification 1: Multi-step expressions are limited to any combination of arithmetic operations, including parentheses, with whole numbers, decimals and fractions.

Clarification 2: Within this benchmark, the expectation is not to include exponents or nested grouping symbols.

Clarification 3: Decimals are limited to hundredths. Expressions cannot include division of a fraction by a fraction.


Examples:
Patti says the expression 12÷2×3 is equivalent to 18 because she works each operation from left to right. Gladys says the expression 12÷2×3 is equivalent to 2 because first multiplies 2×3 then divides 6 into 12. David says that Patti is correctly using order of operations and suggests that if parentheses were added, it would give more clarity.
MA.5.AR.2.3: Determine and explain whether an equation involving any of the four operations is true or false.
Clarifications:
Clarification 1: Problem types include equations that include parenthesis but not nested parentheses.

Clarification 2: Instruction focuses on the connection between properties of equality and order of operations.


Examples:
The equation 2.5+(6×2)=16-1.5 can be determined to be true because the expression on both sides of the equal sign are equivalent to 14.5.
MA.5.AR.2.4: Given a mathematical or real-world context, write an equation involving any of the four operations to determine the unknown whole number with the unknown in any position.
Clarifications:
Clarification 1: Instruction extends the development of algebraic thinking where the unknown letter is recognized as a variable.

Clarification 2: Problems include the unknown and different operations on either side of the equal sign


Examples:
The equation 250-(5×s)=15 can be used to represent that 5 sheets of paper are given to s students from a pack of paper containing 250 sheets with 15 sheets left over.
MA.5.AR.3.1: Given a numerical pattern, identify and write a rule that can describe the pattern as an expression.
Clarifications:
Clarification 1: Rules are limited to one or two operations using whole numbers.

Examples:
The given pattern 6,8,10,12… can be describe using the expression 4+2x, where x=1,2,3,4… ; the expression 6+2x, where x=0,1,2,3… or the expression 2x, where x=3,4,5,6….
MA.5.AR.3.2: Given a rule for a numerical pattern, use a two-column table to record the inputs and outputs.
Clarifications:
Clarification 1: Instruction builds a foundation for proportional and linear relationships in later grades.

Clarification 2: Rules are limited to one or two operations using whole numbers.


Examples:

The expression 6+2x, where x represents any whole number, can be represented in a two-column table as shown below.

Input (X)0123
Output 681012

MA.5.DP.1.1: Collect and represent numerical data, including fractional and decimal values, using tables, line graphs or line plots.
Clarifications:
Clarification 1: Within this benchmark, the expectation is for an estimation of fractional and decimal heights on line graphs.

Clarification 2: Decimal values are limited to hundredths. Denominators are limited to 1, 2, 3 and 4. Fractions can be greater than one.


Examples:
Gloria is keeping track of her money every week. She starts with $10.00, after one week she has $7.50, after two weeks she has $12.00 and after three weeks she has $6.25. Represent the amount of money she has using a line graph.
MA.5.DP.1.2: Interpret numerical data, with whole-number values, represented with tables or line plots by determining the mean, mode, median or range.
Clarifications:
Clarification 1: Instruction includes interpreting the mean in real-world problems as a leveling out, a balance point or an equal share.

Examples:
Rain was collected and measured daily to the nearest inch for the past week. The recorded amounts are 1,0,3,1,0,0 and 1. The range is 3 inches, the modes are 0 and 1 inches and the mean value can be determined as begin mathsize 12px style fraction numerator open parentheses 1 plus 0 plus 3 plus 1 plus 0 plus 0 plus 1 close parentheses over denominator 7 end fraction end style which is equivalent to begin mathsize 12px style 6 over 7 end style of an inch. This mean would be the same if it rained begin mathsize 12px style 6 over 7 end style of an inch each day.
MA.5.FR.1.1: Given a mathematical or real-world problem, represent the division of two whole numbers as a fraction.
Clarifications:
Clarification 1: Instruction includes making a connection between fractions and division by understanding that fractions can also represent division of a numerator by a denominator.

Clarification 2: Within this benchmark, the expectation is not to simplify or use lowest terms. 

Clarification 3: Fractions can include fractions greater than one.


Examples:
At Shawn’s birthday party, a two-gallon container of lemonade is shared equally among 20 friends. Each friend will have begin mathsize 12px style 2 over 20 end style of a gallon of lemonade which is equivalent to one-tenth of a gallon which is a little more than 12 ounces.
MA.5.FR.2.1: Add and subtract fractions with unlike denominators, including mixed numbers and fractions greater than 1, with procedural reliability.
Clarifications:
Clarification 1: Instruction includes the use of estimation, manipulatives, drawings or the properties of operations.

Clarification 2: Instruction builds on the understanding from previous grades of factors up to 12 and their multiples.


Examples:
The sum of begin mathsize 12px style 1 over 12 end style and begin mathsize 12px style 1 over 24 end style can be determined as begin mathsize 12px style 1 over 8 end style,begin mathsize 12px style 3 over 24 end style, begin mathsize 12px style 6 over 48 end style or begin mathsize 12px style 36 over 288 end style by using different common denominators or equivalent fractions.
MA.5.FR.2.2: Extend previous understanding of multiplication to multiply a fraction by a fraction, including mixed numbers and fractions greater than 1, with procedural reliability.
Clarifications:
Clarification 1: Instruction includes the use of manipulatives, drawings or the properties of operations.

Clarification 2: Denominators limited to whole numbers up to 20.

MA.5.FR.2.3: When multiplying a given number by a fraction less than 1 or a fraction greater than 1, predict and explain the relative size of the product to the given number without calculating.
Clarifications:
Clarification 1: Instruction focuses on the connection to decimals, estimation and assessing the reasonableness of an answer.
MA.5.FR.2.4: Extend previous understanding of division to explore the division of a unit fraction by a whole number and a whole number by a unit fraction.
Clarifications:
Clarification 1: Instruction includes the use of manipulatives, drawings or the properties of operations.

Clarification 2: Refer to Situations Involving Operations with Numbers (Appendix A).

MA.5.GR.1.1: Classify triangles or quadrilaterals into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category.
Clarifications:
Clarification 1: Triangles include scalene, isosceles, equilateral, acute, obtuse and right; quadrilaterals include parallelograms, rhombi, rectangles, squares and trapezoids.
MA.5.GR.1.2: Identify and classify three-dimensional figures into categories based on their defining attributes. Figures are limited to right pyramids, right prisms, right circular cylinders, right circular cones and spheres.
Clarifications:
Clarification 1: Defining attributes include the number and shape of faces, number and shape of bases, whether or not there is an apex, curved or straight edges and curved or flat faces.
MA.5.GR.2.1: Find the perimeter and area of a rectangle with fractional or decimal side lengths using visual models and formulas.
Clarifications:
Clarification 1: Instruction includes finding the area of a rectangle with fractional side lengths by tiling it with squares having unit fraction side lengths and showing that the area is the same as would be found by multiplying the side lengths.

Clarification 2: Responses include the appropriate units in word form.

MA.5.GR.3.1: Explore volume as an attribute of three-dimensional figures by packing them with unit cubes without gaps. Find the volume of a right rectangular prism with whole-number side lengths by counting unit cubes.
Clarifications:
Clarification 1: Instruction emphasizes the conceptual understanding that volume is an attribute that can be measured for a three-dimensional figure. The measurement unit for volume is the volume of a unit cube, which is a cube with edge length of 1 unit.
MA.5.GR.3.2: Find the volume of a right rectangular prism with whole-number side lengths using a visual model and a formula.
Clarifications:
Clarification 1: Instruction includes finding the volume of right rectangular prisms by packing the figure with unit cubes, using a visual model or applying a multiplication formula.

Clarification 2: Right rectangular prisms cannot exceed two-digit edge lengths and responses include the appropriate units in word form.

MA.5.GR.3.3: Solve real-world problems involving the volume of right rectangular prisms, including problems with an unknown edge length, with whole-number edge lengths using a visual model or a formula. Write an equation with a variable for the unknown to represent the problem.
Clarifications:
Clarification 1: Instruction progresses from right rectangular prisms to composite figures composed of right rectangular prisms.

Clarification 2: When finding the volume of composite figures composed of right rectangular prisms, recognize volume as additive by adding the volume of non-overlapping parts.

Clarification 3: Responses include the appropriate units in word form.


Examples:
A hydroponic box, which is a rectangular prism, is used to grow a garden in wastewater rather than soil. It has a base of 2 feet by 3 feet. If the volume of the box is 12 cubic feet, what would be the depth of the box?
MA.5.GR.4.1: Identify the origin and axes in the coordinate system. Plot and label ordered pairs in the first quadrant of the coordinate plane.
Clarifications:
Clarification 1: Instruction includes the connection between two-column tables and coordinates on a coordinate plane.

Clarification 2: Instruction focuses on the connection of the number line to the x- and y-axis.

Clarification 3: Coordinate planes include axes scaled by whole numbers. Ordered pairs contain only whole numbers.

MA.5.GR.4.2: Represent mathematical and real-world problems by plotting points in the first quadrant of the coordinate plane and interpret coordinate values of points in the context of the situation.
Clarifications:
Clarification 1: Coordinate planes include axes scaled by whole numbers. Ordered pairs contain only whole numbers.

Examples:
For Kevin’s science fair project, he is growing plants with different soils. He plotted the point (5,7) for one of his plants to indicate that the plant grew 7 inches by the end of week 5.
MA.5.M.1.1: Solve multi-step real-world problems that involve converting measurement units to equivalent measurements within a single system of measurement.
Clarifications:
Clarification 1: Within the benchmark, the expectation is not to memorize the conversions.

Clarification 2: Conversions include length, time, volume and capacity represented as whole numbers, fractions and decimals.


Examples:
There are 60 minutes in 1 hour, 24 hours in 1 day and 7 days in 1 week. So, there are 60×24×7 minutes in one week which is equivalent to 10,080 minutes.
MA.5.M.2.1: Solve multi-step real-world problems involving money using decimal notation.
Examples:
Don is at the store and wants to buy soda. Which option would be cheaper: buying one 24-ounce can of soda for $1.39 or buying two 12-ounce cans of soda for 69¢ each?
MA.5.NSO.1.1: Express how the value of a digit in a multi-digit number with decimals to the thousandths changes if the digit moves one or more places to the left or right.
MA.5.NSO.1.2: Read and write multi-digit numbers with decimals to the thousandths using standard form, word form and expanded form.
Examples:
The number sixty-seven and three hundredths written in standard form is 67.03 and in expanded form is 60+7+0.03 or begin mathsize 12px style open parentheses 6 cross times 10 close parentheses plus open parentheses 7 cross times 1 close parentheses plus open parentheses 3 cross times 1 over 100 close parentheses end style.
MA.5.NSO.1.3: Compose and decompose multi-digit numbers with decimals to the thousandths in multiple ways using the values of the digits in each place. Demonstrate the compositions or decompositions using objects, drawings and expressions or equations.
Examples:
The number 20.107 can be expressed as 2 tens + 1 tenth+7 thousandths or as 20 ones + 107 thousandths.
MA.5.NSO.1.4: Plot, order and compare multi-digit numbers with decimals up to the thousandths.
Clarifications:
Clarification 1: When comparing numbers, instruction includes using an appropriately scaled number line and using place values of digits.

Clarification 2: Scaled number lines must be provided and can be a representation of any range of numbers. 

Clarification 3: Within this benchmark, the expectation is to use symbols (<, > or =).


Examples:
Example: The numbers 4.891; 4.918 and 4.198 can be arranged in ascending order as 4.198; 4.891 and 4.918.

Example: 0.15<0.2 because fifteen hundredths is less than twenty hundredths, which is the same as two tenths.

MA.5.NSO.1.5: Round multi-digit numbers with decimals to the thousandths to the nearest hundredth, tenth or whole number.
Examples:
The number 18.507 rounded to the nearest tenth is 18.5 and to the nearest hundredth is 18.51.
MA.5.NSO.2.1: Multiply multi-digit whole numbers including using a standard algorithm with procedural fluency.
MA.5.NSO.2.2: Divide multi-digit whole numbers, up to five digits by two digits, including using a standard algorithm with procedural fluency. Represent remainders as fractions.
Clarifications:
Clarification 1: Within this benchmark, the expectation is not to use simplest form for fractions.

Examples:

The quotient 27÷7 gives 3 with remainder 6 which can be expressed as

begin mathsize 12px style 3 6 over 7 end style.
MA.5.NSO.2.3: Add and subtract multi-digit numbers with decimals to the thousandths, including using a standard algorithm with procedural fluency.
MA.5.NSO.2.4: Explore the multiplication and division of multi-digit numbers with decimals to the hundredths using estimation, rounding and place value.
Clarifications:
Clarification 1: Estimating quotients builds the foundation for division using a standard algorithm.

Clarification 2: Instruction includes the use of models based on place value and the properties of operations.


Examples:
The quotient of 23 and 0.42 can be estimated as a little bigger than 46 because 0.42 is less than one-half and 23 times 2 is 46.
MA.5.NSO.2.5: Multiply and divide a multi-digit number with decimals to the tenths by one-tenth and one-hundredth with procedural reliability.
Clarifications:
Clarification 1: Instruction focuses on the place value of the digit when multiplying or dividing.

Examples:
The number 12.3 divided by 0.01 can be thought of as ?×0.01=12.3 to determine the quotient is 1,230.
MA.K12.MTR.1.1: Actively participate in effortful learning both individually and collectively.  

Mathematicians who participate in effortful learning both individually and with others: 

  • Analyze the problem in a way that makes sense given the task. 
  • Ask questions that will help with solving the task. 
  • Build perseverance by modifying methods as needed while solving a challenging task. 
  • Stay engaged and maintain a positive mindset when working to solve tasks. 
  • Help and support each other when attempting a new method or approach.

 

Clarifications:
Teachers who encourage students to participate actively in effortful learning both individually and with others:
  • Cultivate a community of growth mindset learners. 
  • Foster perseverance in students by choosing tasks that are challenging. 
  • Develop students’ ability to analyze and problem solve. 
  • Recognize students’ effort when solving challenging problems.
MA.K12.MTR.2.1: Demonstrate understanding by representing problems in multiple ways.  

Mathematicians who demonstrate understanding by representing problems in multiple ways:  

  • Build understanding through modeling and using manipulatives.
  • Represent solutions to problems in multiple ways using objects, drawings, tables, graphs and equations.
  • Progress from modeling problems with objects and drawings to using algorithms and equations.
  • Express connections between concepts and representations.
  • Choose a representation based on the given context or purpose.
Clarifications:
Teachers who encourage students to demonstrate understanding by representing problems in multiple ways: 
  • Help students make connections between concepts and representations.
  • Provide opportunities for students to use manipulatives when investigating concepts.
  • Guide students from concrete to pictorial to abstract representations as understanding progresses.
  • Show students that various representations can have different purposes and can be useful in different situations. 
MA.K12.MTR.3.1: Complete tasks with mathematical fluency. 

Mathematicians who complete tasks with mathematical fluency:

  • Select efficient and appropriate methods for solving problems within the given context.
  • Maintain flexibility and accuracy while performing procedures and mental calculations.
  • Complete tasks accurately and with confidence.
  • Adapt procedures to apply them to a new context.
  • Use feedback to improve efficiency when performing calculations. 
Clarifications:
Teachers who encourage students to complete tasks with mathematical fluency:
  • Provide students with the flexibility to solve problems by selecting a procedure that allows them to solve efficiently and accurately.
  • Offer multiple opportunities for students to practice efficient and generalizable methods.
  • Provide opportunities for students to reflect on the method they used and determine if a more efficient method could have been used. 
MA.K12.MTR.4.1: Engage in discussions that reflect on the mathematical thinking of self and others. 

Mathematicians who engage in discussions that reflect on the mathematical thinking of self and others:

  • Communicate mathematical ideas, vocabulary and methods effectively.
  • Analyze the mathematical thinking of others.
  • Compare the efficiency of a method to those expressed by others.
  • Recognize errors and suggest how to correctly solve the task.
  • Justify results by explaining methods and processes.
  • Construct possible arguments based on evidence. 
Clarifications:
Teachers who encourage students to engage in discussions that reflect on the mathematical thinking of self and others:
  • Establish a culture in which students ask questions of the teacher and their peers, and error is an opportunity for learning.
  • Create opportunities for students to discuss their thinking with peers.
  • Select, sequence and present student work to advance and deepen understanding of correct and increasingly efficient methods.
  • Develop students’ ability to justify methods and compare their responses to the responses of their peers. 
MA.K12.MTR.5.1: Use patterns and structure to help understand and connect mathematical concepts. 

Mathematicians who use patterns and structure to help understand and connect mathematical concepts:

  • Focus on relevant details within a problem.
  • Create plans and procedures to logically order events, steps or ideas to solve problems.
  • Decompose a complex problem into manageable parts.
  • Relate previously learned concepts to new concepts.
  • Look for similarities among problems.
  • Connect solutions of problems to more complicated large-scale situations. 
Clarifications:
Teachers who encourage students to use patterns and structure to help understand and connect mathematical concepts:
  • Help students recognize the patterns in the world around them and connect these patterns to mathematical concepts.
  • Support students to develop generalizations based on the similarities found among problems.
  • Provide opportunities for students to create plans and procedures to solve problems.
  • Develop students’ ability to construct relationships between their current understanding and more sophisticated ways of thinking.
MA.K12.MTR.6.1: Assess the reasonableness of solutions. 

Mathematicians who assess the reasonableness of solutions: 

  • Estimate to discover possible solutions.
  • Use benchmark quantities to determine if a solution makes sense.
  • Check calculations when solving problems.
  • Verify possible solutions by explaining the methods used.
  • Evaluate results based on the given context. 
Clarifications:
Teachers who encourage students to assess the reasonableness of solutions:
  • Have students estimate or predict solutions prior to solving.
  • Prompt students to continually ask, “Does this solution make sense? How do you know?”
  • Reinforce that students check their work as they progress within and after a task.
  • Strengthen students’ ability to verify solutions through justifications. 
MA.K12.MTR.7.1: Apply mathematics to real-world contexts. 

Mathematicians who apply mathematics to real-world contexts:

  • Connect mathematical concepts to everyday experiences.
  • Use models and methods to understand, represent and solve problems.
  • Perform investigations to gather data or determine if a method is appropriate. • Redesign models and methods to improve accuracy or efficiency. 
Clarifications:
Teachers who encourage students to apply mathematics to real-world contexts:
  • Provide opportunities for students to create models, both concrete and abstract, and perform investigations.
  • Challenge students to question the accuracy of their models and methods.
  • Support students as they validate conclusions by comparing them to the given situation.
  • Indicate how various concepts can be applied to other disciplines.
ELA.K12.EE.1.1: Cite evidence to explain and justify reasoning.
Clarifications:
K-1 Students include textual evidence in their oral communication with guidance and support from adults. The evidence can consist of details from the text without naming the text. During 1st grade, students learn how to incorporate the evidence in their writing.

2-3 Students include relevant textual evidence in their written and oral communication. Students should name the text when they refer to it. In 3rd grade, students should use a combination of direct and indirect citations.

4-5 Students continue with previous skills and reference comments made by speakers and peers. Students cite texts that they’ve directly quoted, paraphrased, or used for information. When writing, students will use the form of citation dictated by the instructor or the style guide referenced by the instructor. 

6-8 Students continue with previous skills and use a style guide to create a proper citation.

9-12 Students continue with previous skills and should be aware of existing style guides and the ways in which they differ.

ELA.K12.EE.2.1: Read and comprehend grade-level complex texts proficiently.
Clarifications:
See Text Complexity for grade-level complexity bands and a text complexity rubric.
ELA.K12.EE.3.1: Make inferences to support comprehension.
Clarifications:
Students will make inferences before the words infer or inference are introduced. Kindergarten students will answer questions like “Why is the girl smiling?” or make predictions about what will happen based on the title page. Students will use the terms and apply them in 2nd grade and beyond.
ELA.K12.EE.4.1: Use appropriate collaborative techniques and active listening skills when engaging in discussions in a variety of situations.
Clarifications:
In kindergarten, students learn to listen to one another respectfully.

In grades 1-2, students build upon these skills by justifying what they are thinking. For example: “I think ________ because _______.” The collaborative conversations are becoming academic conversations.

In grades 3-12, students engage in academic conversations discussing claims and justifying their reasoning, refining and applying skills. Students build on ideas, propel the conversation, and support claims and counterclaims with evidence.

ELA.K12.EE.5.1: Use the accepted rules governing a specific format to create quality work.
Clarifications:
Students will incorporate skills learned into work products to produce quality work. For students to incorporate these skills appropriately, they must receive instruction. A 3rd grade student creating a poster board display must have instruction in how to effectively present information to do quality work.
ELA.K12.EE.6.1: Use appropriate voice and tone when speaking or writing.
Clarifications:
In kindergarten and 1st grade, students learn the difference between formal and informal language. For example, the way we talk to our friends differs from the way we speak to adults. In 2nd grade and beyond, students practice appropriate social and academic language to discuss texts.
ELD.K12.ELL.MA.1: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.



General Course Information and Notes

VERSION DESCRIPTION

This course supports students who need additional instruction in foundational mathematics skills as it relates to core instruction. Instruction will use explicit, systematic, and sequential approaches to mathematics instruction addressing all domains including number sense & operations, fractions, algebraic reasoning, geometric reasoning, measurement and data analysis & probability. Teachers will use the listed standards that correspond to each students’ needs. 

Effective instruction matches instruction to the need of the students in the group and provides multiple opportunities to practice the skill and receive feedback. The additional time allotted for this course is in addition to core instruction. The intervention includes materials and strategies designed to supplement core instruction.


GENERAL NOTES

Florida’s Benchmarks for Excellent Student Thinking (B.E.S.T.) Standards
This course includes Florida’s B.E.S.T. ELA Expectations (EE) and Mathematical Thinking and Reasoning Standards (MTRs) for students. Florida educators should intentionally embed these standards within the content and their instruction as applicable. For guidance on the implementation of the EEs and MTRs, please visit https://www.cpalms.org/Standards/BEST_Standards.aspx and select the appropriate B.E.S.T. Standards package.

English Language Development ELD Standards Special Notes Section:
Teachers are required to provide listening, speaking, reading and writing instruction that allows English language learners (ELL) to communicate information, ideas and concepts for academic success in the content area of Mathematics. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade-level words, expressions, sentences and discourse to process or produce language necessary for academic success. The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum developers and teachers which maximizes an ELL’s need for communication and social skills. To access an ELL supporting document which delineates performance definitions and descriptors, please click on the following link:  https://cpalmsmediaprod.blob.core.windows.net/uploads/docs/standards/eld/ma.pdf


General Information

Course Number: 5012015 Course Path: Section: Grades PreK to 12 Education Courses > Grade Group: Grades PreK to 5 Education Courses > Subject: Mathematics > SubSubject: General Mathematics >
Abbreviated Title: FDN SKILLS MATH 3-5
Course Attributes:
  • Class Size Core Required
  • Florida Standards Course
Course Type: Elective Course Course Level: 1
Course Status: State Board Approved
Grade Level(s): 3,4,5



Educator Certifications

Elementary Education (Elementary Grades 1-6)
Elementary Education (Grades K-6)
Mathematics (Elementary Grades 1-6)
Middle Grades Mathematics (Middle Grades 5-9)


State Adopted Instructional Materials

STEMscopes Florida Math
Dr. Jarrett Reid Whitaker - Accelerate Learning - First Edition - 2022
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There are more than 2372 related instructional/educational resources available for this on CPALMS. Click on the following link to access them: https://www.cpalms.org/PreviewCourse/Preview/20399