Understanding the Learning Trajectory

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Big Ideas:

• A ratio is a multiplicative comparison of quantities; there are different types of comparisons that can be represented as ratios (Charles, 2005).

• Ratios give the relative sizes of the quantities being compared, not necessarily the actual sizes (Charles, 2005).

• The student understands that relationships between quantities can be part to whole or part to part.

• The student understands that ratios can be written in multiple forms (Common Core Writing Team, 2019, Ratios and Proportional Relationships p. 3).

Important Assessment Look Fors:

• The student correctly determines if the scenario is a part to part or part to whole relationship.

• The student correctly expresses equivalent ratios for the given situation.

• The student expresses ratios in correct order (ex: 1:3 vs 3:1).

• The student expresses ratios using the correct format (a to b, a:b, or a/b).

Purposeful Questions: 

•  How do you know if you are looking for a part to part or a part to whole relationship?

•  What strategies can you use to determine if ratios are equivalent?

• What are the different ways that you can represent a ratio?

Understanding the Learning Trajectory

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Big Ideas:

• Fractions, decimals, and percents can be used to represent a part to whole relationship.

• Students should begin to identify equivalence among fractions and decimals starting with common fractions such as halves, thirds, fourths and eighths as decimal fractions. For example, using a decimal grid and shading ½ and 50/100= .5. A double number line, decimal grids, and rational numbers wheel (see chap. 16 of Van de Walle text) are useful models to connect decimals and fractions as one moves beyond common fractions to continue the development of fraction-decimal equivalence (Van de Walle et al., 2018). 

• Decimal-fraction-percent equivalents can be named not just through procedures but through sense-making with our base-ten system (e.g. ⅖ = 4/10 =4/100= 0.4=40%). For example, although one can use division with the fraction to find the decimal equivalent, another way is to find an equivalent decimal fraction with 10 or 100 in the denominator to find the decimal and percent equivalent (e.g. 12/50=24/100=0.24=24%)

• Naming an equivalent fraction, decimal and percent means that the quantities are the same even though they are represented differently. ¾ is 0.75 and 75% written in a different form. Both ¾, 0.75, and 75%  would appear at the same point on a number line, take up the same amount of space on an area model, and be shown similarly in a set or measurement model. 

• Percents are used to solve practical problems including sales, data description, and data comparison. 

Important Assessment Look Fors:

• The student recognizes, identifies, and names equivalent fractions, decimals and percents  with concrete or pictorial models.

• The student recognizes, identifies, and names equivalent fractions, decimals and percents without concrete or pictorial models.

• The student demonstrates an understanding that fractional models (such as the one above showing ⅘) can also be written in many equivalent decimal and percent forms (0.8, 0.80, 80% etc.).

• The student can determine the equivalent value given a fraction, percent or decimal.

• The student can justify with reasoning why one form is equivalent to another form representing the same value.

Purposeful Questions: 

• How do you know this model represents the equivalent values between fractions, decimals and percents?

• How does the model help you identify the equivalent values between fractions, decimals and percents?

• How can you compare two percents using pictorial representations?

• What strategy did you use to order your rational numbers?

• Which strategy is the most efficient for you and why? 

Understanding the Learning Progression

Big Ideas:

• Any number can be represented in an infinite number of ways that have the same value and can be compared by their relative values (Charles, p.10, p.14). In order to use reasoning skills when comparing fractions, it is important to have students notice what happens to the size of fractions when the numerator increases and also when the denominator increases. 

• In terms of decimal reasoning, students need to develop the notion that there is what we call decimal density where in between any two decimals there are an infinite number of other decimals (Widjaja et al., 2008).

• Since fractions, decimals and percents are essentially the same numbers in different forms, they can be compared and ordered. Fractions, decimals, and percents can be compared and ordered using a variety of strategies including using benchmarks (0, halves, wholes), naming equivalencies, and other reasoning strategies. 

• Decimal to fraction-to percent equivalents can be named not just through procedures but through sense-making with our base-ten system (e.g. ⅖ = 4/10 = 40/100=40%= 0.4). For example, although one can use division with the fraction to find the decimal equivalent, another way is to find an equivalent decimal fraction with 10 or 100 in the denominator to find the decimal and percent equivalent (e.g. 12/50=24/100=0.24=24%).

Important Assessment Look Fors:

• Students can use a variety of strategies to order and compare rational numbers (close to 0, 1 or ½, converting to a different representation, using a model, using benchmarks, using equivalencies,  etc…) 

• The student demonstrates an understanding of percents that include a decimal point.  

• The student demonstrates understanding of symbols repeating decimals.

• The student can explain the strategy and reasoning used to determine the order of the rational numbers.

• The student can determine a fraction, decimal, or percent that can fit a series of given criteria (less than, greater than, or between two quantities). 

Purposeful Questions: 

• Can you explain to me how you were able to determine that quantity a is less than/greater than/equal to quantity b?

• What strategy/strategies did you use in order to compare/order your numbers? Why was this an effective strategy? 

• How are the strategies you use to compare and order fraction similar or different to the strategies you use to compare decimals and percents? 

• How do you know ___ is greater/less than ___? (fill in rational numbers in the problem)

• Where might you place these numbers on the number line?

• What other way could you represent this rational number?

Understanding the Learning Progression

Big Ideas:

• Integers are the whole numbers and their opposites on the number line, where zero is its own opposite.

• Each integer can be associated with a unique point on the number line, but there are many points on the number line that cannot be named by integers.

• An integer and its opposite are the same distance from zero on the number line. (Charles, 2005)

Important Assessment Look Fors:

• Students can correctly identify a value of a positive or negative integer in relation to 0 on a number line.

• Student has conceptual knowledge or real life situations, such as temperature (above/below zero), deposits/withdrawals in a checking account, golf (above/below par), time lines, football yardage, positive and negative electrical charges, and altitude (above/below sea level), students may not know what some of these are based on their cultural situation.

• Students should be able to model with a number line or with manipulatives to represent the situation.

• Students should be able to determine the opposite number of a given value.

Purposeful Questions: 

• What is the difference in values of negative numbers in relation to 0 on a number line?

• What are some examples of practical situations where an integer can be used?

• What are opposites?

↗Understanding the Learning Progression

Big Ideas:

• An integer and its opposite are the same distance from zero on the number line.

• There is no greatest or least integer on the number line.

• A number to the right of another on the number line is the greater number.

• Numbers can be compared using greater than, less than, or equal.(Charles, 2005)

• Statements of inequality can be interpreted as statements about the relative position of two numbers on a number line diagram. (Common Core Writing Team, 2019, p. 8).

Important Assessment Look-fors: 

• The student understands the meaning of ascending and  descending.

• The student understands mathematical symbols <, > and cen conceptualize them from left to right, and right to left.

• The student understands that the further away from zero a negative number is, the smaller its value.

• The student understands the interval on the number line.

Purposeful questions:

• What intervals are measured on the number line?

• Where do you see  positive and negative values?

• What does it mean when a negative number has a greater distance away from 0 on a number line?

• How do you know this integer is larger?   How do you know this integer is smaller?

Understanding the Learning Progression

Big Ideas:

• Integers are the whole numbers and their opposites on the number line, where zero is its own opposite.

• Each integer can be associated with a unique point on the number line, but there are many points on the number line that cannot be named by integers.

• The distance between an integer and zero on a number line is the integer’s absolute value (Common Core Standards Writing Team, 2019).

• An integer and its opposite are the same distance from zero on the number line (Charles, 2005).

Important Assessment Look-fors: 

• The student recognizes and interprets the symbols for absolute value. 

• The student represents absolute value as positive.

• The student models the absolute value of integers on a number line.

• The student explains that opposite integers have the same absolute value because they are the same distance from zero.

Purposeful questions:

• How do you determine the absolute value?

• How can you show that opposite integers have the same absolute value?

• Why is an absolute value always positive?

Understanding the Learning Progression

Big Ideas:

• A perfect square is a whole number whose square root is an integer.

• Perfect squares can be represented by the area of a square, and the square root of a number can be represented geometrically as the length of a side of the square. 

• Exponents are a way to express repeated products of the same number (Van de Walle).

Important Assessment Look-fors: 

• The student continues a pattern of exponents.

• The student recognizes the relationship between perfect squares and square roots.

• The student uses multiplication facts to find square roots. 

• The student demonstrates an understanding of place value when working with powers of 10

• The student uses repeated multiplication to determine the value of an exponent.

Purposeful questions:

• How can you represent a perfect square pictorially?

• How can you describe the relationship between a perfect square and a square root?

• What is the relationship between the base and the exponent?

Understanding the Learning Progression

Big Ideas:

• Different real-world interpretations can be associated with division calculations involving fractions (decimals).

• The product of two positive fractions each less than one is less than either factor.

• A fraction division calculation can be changed to an equivalent multiplication calculation (i.e., a/b ÷ c/d = a/b x d/c, where b, c, and d = 0) (Charles, 2005).

Important Assessment Look-fors: 

• The student can model multiplication and division with multiple representations.

• The student can use estimation to develop computational strategies.

• The student can demonstrate understanding/justify that when multiplying a fraction by a fraction, this results in the part of a part.

• The student can demonstrate understanding/justify that when multiplying a fraction by a whole number, this results in the part of a whole.

Purposeful questions:

• How can estimation be used to determine the reasonableness of a solution?

• When a whole number is divided by a fraction, why is the quotient larger than the dividend?

• What kind of product do you get when you multiply a whole number by a fraction?

• Can you show me how your multiplication/division model relates to your multiplication/division computation?

Understanding the Learning Progression

Big Ideas:

• The real-world actions for addition and subtraction of whole numbers are the same for operations with fractions.

• Different real-world interpretations can be associated with the product of a whole number and fraction, a fraction and whole number, and a fraction and fraction. 

• Different real-world interpretations can be associated with division calculations involving fractions (Charles, 2005).

• Fractional relationships can be used when solving practical problems involving addition, subtraction, multiplication and/or division (Common Core Writing Team, 2019, Number Systems p. 5-6)

• In mathematics, emphasis should be placed on representing the problem and applying reasoning to understand it rather than relying on keywords.  (See Grade 4 VDOE Standards of Learning Document↗ p.19).

• In mathematics, estimation should be used to determine if an answer is reasonable.

Important Assessment Look-fors: 

• The student can model fraction operations using manipulatives or pictorial representations.

• The student can show their answer expressed in simplest form.

• The student can justify why the number sentence matches the model and situation presented in the problem.

• The student can justify why the answer makes sense.

• The student can give a complete answer, including units of measurement.

Purposeful questions:

• What are you trying to find in the problem? 

• How can you begin to organize your thinking ? Will a picture or chart help you?

• What is happening in the problem? What does that tell you about which operation(s) you will need to use? 

• How do you know your answer is reasonable and what does it mean?

Understanding the Learning Progression

Big Ideas:

• The real-world actions for addition and subtraction of whole numbers are the same for operations with decimals.

• Different real-world interpretations can be associated with the product of a whole number and decimal, a decimal and whole number, and a decimal and decimal. 

• Different real-world interpretations can be associated with division calculations involving decimals (Charles, 2005).

• In mathematics, emphasis should be placed on representing the problem and applying reasoning to understand it rather than relying on keywords  (See Grade 4 VDOE Standards of Learning Document↗ p.19).

• Examples of practical situations solved by using estimation strategies include shopping for groceries, buying school supplies, budgeting an allowance, and sharing the cost of a pizza or the prize money from a contest.

• Different strategies can be used to estimate the result of computations and judge the reasonableness of the result.

Important Assessment Look-fors: 

• The student determines the correct operation or operations needed to solve the problem and can justify his/her choices.

• The student uses estimation to determine reasonableness of solution.

• The student can justify why the answer makes sense.

• The student uses a strategy to organize the information presented in the problem, such as a chart, diagram, list, or picture.

Purposeful questions:

• What are you trying to find in the problem? 

• How can you begin to organize your thinking ? Will a picture or chart help you?

• How do you know your answer is reasonable and what does it mean?

• How can you determine the operation or operations that can be used to solve the problem?

• Have you answered the question asked in the problem?

Understanding the Learning Progression

Big Ideas:

• Integers are the whole numbers and their opposites on the number line, where zero is its own opposite.

• The real-world actions for operations with integers are the same for operations with whole numbers. (Charles, 2005).

• The sum of opposite integers is equal to zero. (Arizona, Number System, p. 7-9)

Important Assessment Look-fors: 

• The student has demonstrated a clear understanding of positive and negative signs in their model.

• The student correctly simplified the expression.

• The student is able to accurately reflect the operation in their model.

• The student can justify the sign they used as part of their solution.

Purposeful questions:

• How do you know that your model represents the expression?

• How can you use the models to show how to simplify the expression?

• How do you know that the answer should be negative?  Positive?  Either?

• How can you use other manipulatives to prove your solution?

• How can you use the number line to show this number sentence?

Understanding the Learning Progression

Big Ideas:

• Numbers can be represented in multiple ways, including scientific and exponential notation, negative numbers and irrational numbers. 

• Integers are negative and positive counting numbers and 0. Positive and negative numbers describe quantities having both magnitude and direction (e.g. temperature above or below zero) (Van De Walle et al., 2018).

• Positive and negative numbers occur in a real world context (temperature, sea level)

• There are multiple meanings of the negative sign: a) a subtraction sign, b) a negative sign, c) an opposite sign

• Properties of operations can be used to make connections between integer operations and whole number operations (Common Core Standards Writing Team, 2019).

Important Assessment Look-fors: 

• The student uses the correct operation to solve the practical situations.

• The student demonstrates understanding of positive and negative movements within the scenario.

• The student models the problem using pictures, words, and/or numbers.

• The student can justify why the solution makes sense within the context of the problem.

Purposeful questions:

• Can you explain what is happening in the problem?

• How might you draw a picture to represent what is happening in the problem?

• How can this scenario be modeled using a number line?

• How can you determine which operation is needed to solve this problem?

• Could you solve this problem in a different way?

Understanding the Learning Progression

Big Ideas:

• An expression is a phrase about a mathematical situation while an equation shows that two expressions are equivalent in value.

• Properties of operations can be used to make connections between integer operations and whole number operations (Common Core Standards Writing Team, 2019).

• Working with numerical expressions prepares students for working with algebraic expressions.

• The order of operations tells us how to interpret expressions (Common Core Standards Writing Team, 2019).

• Integers are the whole numbers and their opposites on the number line, where zero is its own opposite.

• The real-world actions for operations with integers are the same for operations with whole numbers (Charles, 2005).

• The sum of opposite integers is equal to zero (Common Core Standards Writing Team, 2019).

• There are multiple meanings of the negative sign: a) a subtraction sign, b) a negative sign, c) an opposite sign.

Important Assessment Look-fors: 

• The student follows the order of operations when simplifying expressions.

• The student completes multiplication/division or addition/subtraction from left to right.

• The student recognizes parentheses, a fraction bar, and absolute value as grouping symbols.

• The student shows all steps required when simplifying an expression.

• The student can identify errors in simplifying expressions.

Purposeful questions:

• What is your first step in simplifying this expression?

• How does the absolute value symbol affect this term?

• How can you determine if an equation is true or false?

• How might your solution change if you do not follow the order of operations?

Understanding the Learning Progression

Big Ideas:

• A circle is not a polygon because it is not made of straight line segments. The circumference of the circle is made by an infinite number of points that are equidistant from the center of the circle. This distance from the center is called the radius. 

• Proportional relationships exist between the radius, diameter, and circumference, so that given one measurement one can find the magnitude of the others. 

• A diameter is a special type of chord that travels through the center of the circle and is twice the length of the radius and approximately 3 times smaller than the circumference. 

• Pi represents the constant ratio of the circumference to the diameter of any circle (Charles, 2005). Thus, the circumference of a circle is proportional to its diameter (VDOE) and is approximately three times the diameter.

• The area of a circle is the number of complete, non-overlapping square units that fill the inside of the closed curve, and can be found with the formula A = 𝜋r2.

Important Assessment Look-fors: 

• The student uses the ratio of diameter to circumference to approximate pi.

• The student shows understanding of the relationship between the radius, diameter, and circumference.

• The student recognizes the difference between area and circumference of a circle. 

• The student calculates circumference based on a given diameter or radius.

• The student calculates the area of a circle based on a given diameter or radius.

• The student differentiates between situations where they are determining area and situations where they are determining circumference.

Purposeful questions:

• What is the relationship between the radius and the diameter?

• How can we use the diameter to estimate the circumference?

• How do you know if you need to find the circumference or the area?

• What do you need to know about a circle in order to find the circumference/area?

Understanding the Learning Progression

Big Ideas:

• Shapes are classified by the properties they share (Van de Walle et al., 2018).

• Perimeter is the boundary of a two dimensional shape and the area is the space inside that boundary.

• The concept of area being measured in square units can be demonstrated by two-dimensional rectangular regions constructed as rows and columns of squares. 

• Any side of a triangle can be considered the base when calculating area.

• A triangle is half of a parallelogram, and the area of a triangle is half that of its corresponding parallelogram.

Important Assessment Look-fors: 

• The student distinguishes between when they need to find area and when they need to find perimeter.

• The student uses the height of the triangle to find the area.

• The student uses the area/perimeter to find the dimensions of a rectangle.

• The student answers all parts of multi-step questions.

Purposeful questions:

• What is being determined when we find out the amount of fencing needed?

• How do we determine the area of a rectangle?

• How do we determine the perimeter of a shape?

• How do we determine the area of a triangle?

• How can you use the area and perimeter to determine the length of the sides?

• What are the steps needed to solve this problem?

Understanding the Learning Progression

Big Ideas:

• As students develop, locating objects move from spatial directions like under and over, to points on a map and the use of coordinate systems. The first quadrant of the coordinate plane is introduced in grade 5, then in grade 6 all quadrants are included. Later this will build on scale drawing and construction as well as coordinate axis to graph lines, perform transformation and explore distance (Van de Walle et al., 2018).

• The coordinate plane can be used to represent and solve practical problems, and model two-dimensional shapes using the coordinate plane system  (Common Core Standards Writing Team, 2019).

• There are some conventions that students will need to develop, such as, the quadrants of a coordinate plane are the four regions created by the two intersecting perpendicular lines (x- and y-axes). Quadrants are named in counterclockwise order. The signs on the ordered pairs for quadrant I are (+,+); for quadrant II, (–,+); for quadrant III, (–, –); and for quadrant IV, (+,–).Apply the concept of parallel and perpendicular lines to number lines (Grade 6 Standard of Learning Curriculum Framework). 

Important Assessment Look-fors: 

• The student correctly identified the x and y axis.

• The student demonstrates understanding of positive and negative numbers within the coordinate plane.

• The student is familiar with and correctly uses coordinate plane vocabulary including: horizontal, vertical, quadrant, and origin.

• The student correctly identifies the x and y coordinate of an ordered pair.

Purposeful questions:

• How can you determine the quadrant where an ordered pair should be placed?

• What is some information you can include on your coordinate plane to help you locate the correct quadrants?

• What is one way you can help yourself remember the order of the quadrants in the coordinate plane?

• Where would “3” go on the x-axis? Where would “3” go on the y-axis?  Where would -4 go on the y axis?  Where would -4 go on the x axis?

• Why is (0,4) not in a quadrant?  What about (4,0)?  Where would you describe their location?

Understanding the Learning Progression

Big Ideas:

• As students develop, locating objects move from spatial directions like under and over, to points on a map and the use of coordinate systems. The first quadrant of the coordinate plane is introduced in grade 5, then in grade 6 all quadrants are included. Later this will build on scale drawing and construction as well as coordinate axis to graph lines, perform transformation and explore distance (Van de Walle et al., 2018).

• The coordinate plane can be used to represent and solve practical problems, and model two-dimensional shapes using the coordinate plane system  (Common Core Standards Writing Team, 2019).

• There are some conventions that students will need to develop, such as, the quadrants of a coordinate plane are the four regions created by the two intersecting perpendicular lines (x- and y-axes). Quadrants are named in counterclockwise order. The signs on the ordered pairs for quadrant I are (+,+); for quadrant II, (–,+); for quadrant III, (–, –); and for quadrant IV, (+,–). Apply the concept of parallel and perpendicular lines to number lines (Grade 6 Standard of Learning Curriculum Framework). 

• The concept of absolute value can be used to find the distance between two points on the same horizontal or vertical line.

Important Assessment Look-fors: 

• The student has graphed points that fall on the x or y-axis correctly.

• The student has identified points accurately on the coordinate plane. 

• The student can distinguish between the x and y-axis.

• The student can differentiate points that fall on an axis or in a quadrant.

• The student has correctly identified the distance between two points by using a number sentence and absolute values or by modeling the distance on the coordinate plane.

Purposeful questions:

• What technique did you use to help you plot your coordinates correctly?

• What does it mean when one of your coordinates is a 0?

• What method did you use to find the distance between the two points?

• What would happen to the point location if you switched the x and y value?

Understanding the Learning Progression

Big Ideas:

• Geometric properties make shapes alike and different.