UNDERSTANDING THE LEARNING Trajectory

Big Ideas:

• The density property states that between any two real numbers lies another real number. The set of real numbers is all the numbers that have a location on the number line. 

• The real numbers include natural numbers or counting numbers, whole numbers, integers, rational numbers (fractions and repeating or terminating decimals), and irrational numbers. 

• Patterns help us to order and compare real numbers. 

• In mathematics, we can compare and order real numbers by converting between different forms of numbers (ie. fractions → decimals or scientific notation → standard notation).

• There are numbers that are not rational. We can approximate them using rational numbers (Achieve the Core)↗

Important Assessment Look Fors:

• The student can differentiate between negative and positive integers and place them on the correct side of zero.

• The students can convert between scientific notation and decimals.

• The student can use a number line as a tool to correctly order numbers.

• The student can approximate irrational numbers using rational numbers.

Purposeful Questions: 

• What does it mean if a number is negative or positive? How does that affect its placement on the number line?

• What does a negative exponent mean? What does a positive exponent mean?

• When a value isn’t listed on the number line, how do you determine its placement? Does it matter where it goes if there isn’t a given label?

Understanding the Learning Trajectory

Big Ideas:

• Real numbers are involved in everyday life.  They are infinite and can be associated with a unique point on the number line.  

• Real numbers represent a specific quantity.

Important Assessment Look Fors:

• The student can classify values into specific subsets of the real number system

• The student can compare and contrast the different subsets of the real number system

• The student can determine which subset the sum or product of rational and/or irrational numbers belong

• Demonstrate the relationship between the subsets of rational numbers using a venn diagram, table or chart

Purposeful Questions: 

• What is an example proving the sum/product of rational numbers is rational?

• What is an example proving the sum/product of a rational and irrational number is irrational?

• How did you determine which subset(s) this value belongs in?

Understanding the Learning Trajectory

Big Ideas:

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

• The square root of a given number is any number which, when multiplied times itself, equals the given number. Square roots can be positive or negative.

• The square root of a whole number that is not a perfect square is an irrational number (e.g., √2 is an irrational number). An irrational number cannot be expressed exactly as a fraction a/b where b does not equal 0.

• The positive and negative square root of any whole number other than a perfect square lies between two consecutive integers (e.g.,√57 lies between 7 and 8 since 7^2 = 49 and 8^2= 64; √-11 lies between -4 and -3 since (-4)^2 = 16 and (-3)^2  = 9). 

Important Assessment Look Fors:

• Understand the relationship between a square root and its integer value on a number line.

• Understand that there are positive and negative square roots.

• Understand the meaning of the word consecutive when talking about the integers between the two square roots lie.

Purposeful Questions: 

• What numbers could be squared to give you 25? (both +5 and  -5 are solutions)

• What strategy could you use to determine which whole numbers the square root is between? (could make a list of perfect squares).

• If √64 is 8 on the number line, where is √66? (adjust for appropriate numbers)

• Is the √20 closer to 4 or 5 on a number line? Why?

Understanding the Learning Trajectory

Big Ideas:

• Students move from being able to reason proportionally with whole numbers to rational numbers. Students  develop their proportional reasoning while working with ratios, rates, and unit rates representing them with expressions, tape diagrams, double number line diagrams, and tables. They may use these representations to reason about situations involving consumer application (unit price, discounts, tax and tips) as well as color mixtures, recipes, constant speed, and measurement conversions.

• Context clues and vocabulary are used to describe real world situations that can be written using operations with rational numbers.

Important Assessment Look Fors:

• Students understand how to use proportional reasoning to find a percent of a number. 

• Students use vocabulary to determine appropriate operations to perform to solve given consumer scenarios.

• The student should answer the appropriate question. 

Purposeful Questions: 

• Is your answer reasonable for the question? 

• Did you answer the question in its entirety?

• How did you find the commission ( or sales tax, or tip, depending on the question)?

• Would your total decrease or increase after including the sales tax, tip, or commission?

• What is another way you could find the total?

Understanding the Learning Trajectory

Big Ideas:

• Relationships can be found within angles, and those relationships can be described and proved.

• Relationships between angles can be used to solve problems for unknown angles.

Important Assessment Look Fors:

• The student should locate and identify adjacent angles as two non-overlapping angles that share a common ray and a common vertex.

• The student should locate and identify vertical angles as a pair of nonadjacent angles formed by two intersecting lines. Vertical angles are congruent and share a common vertex.

• The student describes complementary angles as any two angles such that the sum of their measures is 90°and explains that adjacent complementary angles form a right angle.

• The student identifies and describes that supplementary angles are any two angles such that the sum of their measures is 180° and form a straight angle.

Purposeful Questions: 

• Are vertical angles equal?

• How are complementary angles different from supplementary angles?  How are they different from vertical angles?

• Explain how you determined the measure of the missing angle.  How can you check your answer?  How do you know your answer is reasonable?

Understanding the Learning Trajectory

Big Ideas:

• Calculating volume and surface area has many applications in real life.

• Understanding the dimensions of a 2D net can be useful in determining surface area and volume of the 3D solid.

• Students learn how these figures are used in the real world and understand the applications of volume and surface area in regards to these figures.

Important Assessment Look Fors:

• Students can determine applications for volume and surface area figures

• Students can calculate volume and surface area for cones given specific dimensions

• Students can calculate volume and surface area for square-based pyramids with given dimensions

Purposeful Questions: 

• What information helped you determine if this problem involved volume or surface area?

• How are the methods for finding the volume of a cone and a square based pyramid similar and different?

Understanding the Learning Trajectory

Big Ideas:

• Volume of a rectangular prism is affected when one dimension is changed through multiplication.  

• The surface area of a rectangular prism is affected when one dimension is multiplied by a factor.

Important Assessment Look Fors:

• When one attribute of a rectangular prism is changed by a specific scale factor, students can determine the effect of the change on the prism’s volume with respect to that scale factor.

• Students can determine a missing attribute by applying the concept of volume and scale factor to a given practical problem.

• Students can determine the difference between the effect a change in one attribute has on the volume of the prism versus the effect on the surface area of that prism.

Purposeful Questions: 

• Are the volumes the same or different when the only dimension changed is the length, width, or height? Why or why not?

• How can drawing a picture of the two prisms help in solving for the missing dimension?

• What strategy did you use to solve for the missing attribute?

Understanding the Learning Trajectory

Big Ideas:

• Transformations are used to identify predictable patterns of how shapes are copied on 2D surfaces.

• Transformations such as translations, rotations, reflections and dilations result in resulting images that are commonly different than the original (or preimage).

Important Assessment Look Fors:

• The student can identify the coordinates of an image after a vertical and/or horizontal translation, a reflection over the x-axis or y-axis.  (8.7a)

• The student can sketch a reflection followed by a translation or a translation followed by a reflection.  The student can determine the coordinates of the image after performing the series of transformations.  (8.7a)

• The student can sketch an image after a dilation about the origin by a specific scale factor. (8.7a)

• The student can correctly identify a performed transformation or series of transformations given a preimage and image. (8.7b)

Purposeful Questions: 

• How did you decide where to place your image?

• What stayed the same and what changed from your preimage to your image

• Was the order in which the transformations were applied important?  Why?

• How did the sketch help in determining the type of transformation(s)?

Understanding the Learning Trajectory

Big Ideas:

• Constructing three-dimensional models from differing views builds spatial reasoning.

• Spatial reasoning provides perspective in real world context. 

Important Assessment Look Fors:

• The student correctly identified two-dimensional view points from a three-dimensional drawing.

• The student correctly drew a three-dimensional figure given a front, top and side view.

• The student correctly drew a front, top or side view from a given three-dimensional image.  

Purposeful Questions: 

• What can you say about the dimensions of your shape?

• What did you use to determine the front, top, or side view?

• What did you look for when creating the three-dimensional image?

• How does identifying different perspectives apply to real world situations?

Understanding the Learning Trajectory

Big Ideas:

• The Pythagorean Theorem is used to determine the measure of any one of the three sides of a right triangle if the measures of the other two sides are known.

• The converse of the Pythagorean Theorem states that if the square of the length of the hypotenuse equals the sum of the squares of the legs in a triangle, then the triangle is a right triangle. This can be used to determine whether a triangle is a right triangle given the measures of its three sides. 

• The hypotenuse of a right triangle is the side opposite the right angle.

• The hypotenuse of a right triangle is always the longest side of the right triangle.

• The legs of a right triangle form the right angle

Important Assessment Look Fors:

• Students understand the relationships between the sides of a right triangle are characterized by the equation a^2+b^2=c^2. 

• Students understand that the legs of the right triangle are sides a and b and the hypotenuse is side c.

• Students can use area models to prove the Pythagorean Theorem

Purposeful Questions: 

• What is the relationship between the legs of a right triangle and the right angle?

• Where is the hypotenuse of a right triangle always located?

Understanding the Learning Trajectory

Big Ideas:

• Lengths of triangle and square sides can be compared using ideas such as longer, shorter, and equal (Charles, 2005).

• Behind every measurement formula lies a geometric result (NCTM)

• Triangles and quadrilaterals can be described, categorized, and named based on the relative lengths of their sides and the sizes of their angles (Charles, 2005). 

Important Assessment Look Fors:

• The student can square numbers and find square roots.

• The student can identify the hypotenuse of a triangle.

• The student can identify right triangles given side lengths by using the Pythagorean Theorem.

• The student can find the length of the hypotenuse or the legs using the Pythagorean theorem.

• The student can describe how to use the Pythagorean Theorem when applied to real-world problems. 

Purposeful Questions: 

• When thinking about the Pythagorean Theorem, does it matter which side length we substitute for each variable

• (a2+b2 = c2)? Why or why not?

• What is the hypotenuse? How is it relevant to real world applications?

• What is distance? How do we measure it? Are there different ways of measuring distance when thinking about real-world applications?

• How did you arrive at your length for the hypotenuse?  Explain how you used the Pythagorean Theorem.

Understanding the Learning Trajectory

Big Ideas:

• Polygons can be described uniquely by their sides and angles. (Charles, 2005)

• Polygons can be constructed from or decomposed into other polygons. (Charles, 2005)

• Behind every measurement formula lies a geometric result.

Important Assessment Look Fors:

• The student can calculate the area and perimeter of triangles, rectangles, squares, trapezoids, parallelograms, and semicircles.

• The student can use composition and/or decomposition to find the area or perimeter of a composite plane figure.

• The student applies vocabulary associated with each shape (ie. parts of a circle, what makes a trapezoid, etc.) to correctly calculate the perimeter or area of the parts of the composite plane figures using the appropriate formulas.

Purposeful Questions: 

• How do we find the lengths of the unknown sides? Can we find the area or perimeter without finding these missing side lengths first?

• Can we assume two side lengths are congruent if they look the same? Why or why not?

• What is the formula for finding the area of this shape? Why does that formula make sense?

• How did you arrive at your final area?  Explain the process you used.  Could you have found it a different way?

Understanding the Learning Trajectory

Big Ideas:

• Probability can provide a basis for making predictions. (Charles, 2005)

• Some probabilities can be determined through experimental trials. (Charles, 2005)

• The chance of an event occurring can be described numerically by a number between 0 and 1 inclusive and used to make predictions about other events. (Charles, 2005))

• Two events are either dependent or independent.  If the outcome of one event has an impact on the outcome of the other event, the events are called dependent. If events are dependent then the second event is considered only if the first event has already occurred. (VDOE Curriculum Framework)

Important Assessment Look Fors:

• The student can determine if two events are independent or dependent and explain their reasoning.

• The student can explain how the experimental probability changes in different scenarios (ie. if an item is replaced vs. if an item is not replaced).

• The student can compare probabilities in the form of rational numbers in order to identify which event has a higher or lower probability. 

Purposeful Questions: 

• What is the difference between an independent and dependent event?

• Why are these two events independent?  Dependent?  Explain your reasoning.

• How does the act of replacing or not replacing an object affect probability?

Understanding the Learning Trajectory

Big Ideas:

• Probability can provide a basis for making predictions. (Charles, 2005)

• Some probabilities can be determined through experimental trials. (Charles, 2005)

• The chance of an event occurring can be described numerically by a number between 0 and 1 inclusive and used to make predictions about other events. (Charles, 2005))

• Two events are either dependent or independent.  If the outcome of one event has an impact on the outcome of the other event, the events are called dependent. If events are dependent then the second event is considered only if the first event has already occurred. (VDOE Curriculum Framework)

Important Assessment Look Fors:

• The student can define the difference between independent and dependent events. 

• The student can explain how the theoretical and experimental probability changes based on if the events occuring are independent or dependent. 

• The student can compare probabilities in the form of rational numbers in order to identify which event has a higher or lower probability.

Purposeful Questions: 

• How many events are occurring? One or more? Are the events independent or dependent? How do you know?

• How do we calculate the probability of dependent events? Independent events?

• When calculating the probability of an event and writing it as a fraction - what value goes in the numerator? Why? What value goes in the denominator? Why? 

• How would the probability change if the item IS replaced?  Or IS NOT replaced?  Explain how you know.

Understanding the Learning Trajectory

Big Ideas:

• A boxplot is an effective way to organize univariable data

• Boxplots provide statistical information including median and range.

• Quartiles (or 25%) are used in boxplots to show a visualization of means and medians of the data.

• When given two sets of data, boxplots are a helpful method for making comparisons. 

Important Assessment Look Fors:

• The student did not include the median when calculating the lower quartile and upper quartile when given a odd number of data values (8.12a)

• The student correctly identified all 5 of the critical points when constructing a boxplot (8.12a)

• The student correctly identified data from a given boxplot (8.12b)

• The student understands how a boxplot organizes data by equal quartiles or percentages (8.12b)

• The student can make comparisons between two boxplots regarding interquartile ranges, percentages and median values. (8.12c)

Purposeful Questions: 

• How does putting the data in numerical order help in determining the 5 critical points of the boxplot? 

• Given this incorrect statement, how can you change the statement to make it true?

• What is similar and what is different between the right and left whiskers?

• If another piece of data was added to this box plot, how would that affect the graph? 

• How can these boxplots be useful for real world application?

Understanding the Learning Trajectory

Big Ideas:

• Scatterplots are effective ways to organize two-variable data.

• Scatterplots can be used to observe trends and make predictions.

• Students can determine positive, negative or no relationship between two sets of data displayed in a scatterplot

Important Assessment Look Fors:

• The student plotted the independent variable values on the x-axis and the corresponding dependent variable values on the y-axis. (8.13a)

• The student can correctly record data from a scatterplot and plot ordered pairs on a coordinate plane to form a scatterplot when the data is written in whole number or half number values. (8.13a)

• The student can interpret positive, negative or no correlations correctly from a given scatterplot. (8.13b)

• The student can make connections between the correlation of a scatterplot and the situation modeled in the graph.  (8.13b)  

Purposeful Questions: 

• What does this value in the scatterplot represent?

• Describe the relationship between the independent variable and the dependent variable.  How is the independent variable causing a change in the dependent variable?

• Given this scatterplot, what situation could be modeled here?

• Given this situation and scatterplot, does this trend imply causation?  Do the x-values really make an impact on the y-values, why or why not?

Understanding the Learning Trajectory

Big Ideas:

• Linear graphs are used to represent relationships between two quantities.

• Lines of best fit are effective tools to make predictions when data fits a linear pattern

• Estimating a line of best fit allows the students to make inferences regarding the patterns and the trends displayed in the scatterplot.

Important Assessment Look Fors:

• The student can identify the line of best fit by judging the line’s closeness to the points in the scatterplot.

• The student can use the line of best fit to make predictions and estimations for a given set of data.

• The student can use the line of best fit to interpret the situation modeled by the data.

Purposeful Questions: 

• How do you know that your line of best fit is accurate or reasonable?

• What does the slope of your line of best fit tell you about the situation modeled?

• How can you determine if your prediction is reasonable?  

• How does your estimation relate to the situation modeled?

Understanding the Learning Trajectory

Big Ideas:

• In mathematics,  it is understood that a variable can be replaced by a value. 

• Letters are used in mathematics to represent generalized properties, unknowns in equations, and relationships between quantities (Charles, 2005).

• In mathematics, following the order of operations is the correct way to simplify/evaluate an expression and there are specific notations to follow.

• In mathematics, performing any operation involving rational numbers is necessary to simplify expressions.

Important Assessment Look Fors:

• Students should be able to replace the variables with the assigned value with appropriate signs.

• Students should start at the innermost grouping symbol in this expression. 

• Students should be able to follow the order of operations accurately with expressions including rational values. 

Purposeful Questions: 

• What are the different types of grouping symbols you see in this expression?

• How do you know where to start? 

• Should the answer be negative or positive? How do you know?

• Explain how you arrived at your final expression.

Understanding the Learning Trajectory

Big Ideas:

• Letters are used in mathematics to represent generalized properties, unknowns in equations, and relationships between quantities (Charles, 2005).

• Algebraic expressions can be named in an infinite number of different but equivalent ways (Charles, 2005).

• Properties of whole numbers apply to certain operations but not others (e.g., The commutative property applies to addition and multiplication but not subtraction and division) (Charles, 2005).

Important Assessment Look Fors:

• Students should correctly use properties of real numbers to simplify expressions.

• Students should properly distribute positive and negative signs, as well as when multiplying a value by an expression. 

• All like terms should be combined in the expression’s simplest form.

Purposeful Questions: 

• Can you identify all the like terms?

• Can your expression be simplified further?

• According to the order of operations and properties of real numbers, what should you do first?

Understanding the Learning Trajectory

Big Ideas:

• Ordered pairs, a table, or a graph of discrete points is a function with the property that each input is related to exactly one output.

• Mathematical rules (relations) can be used to assign members of one set to members of another set. (Charles, 2005)

• A special rule (function) assigns each member of one set to a unique member of the other set. (Charles, 2005)

Important Assessment Look Fors:

• The student can rule out tables and ordered pairs that repeat the x or input value for two separate y or output values.

• The student can identify functions even if the y values are repeated as long as the x value is not.

• The student can recognize that a graph of a function cannot have more than one coordinate plotted on the same vertical line and explain why this “vertical line” test helps to identify functions.

Purposeful Questions: 

• Is this a function?  Explain your reasoning.

• Do any of the x-values repeat in your table or ordered pairs?

• Why can the x-values not repeat? 

• Why can the y-values not repeat?

• Is there only one coordinate on each vertical line representing the x-values for your function?

• How could you change this table, set of ordered pairs, or graph so that it becomes a function?