Understanding the Learning Trajectory

Big Ideas:

• Patterns are found in powers of 10.

Important Assessment Look Fors:

• The student can represent negative powers of ten in decimal and fraction form

• The student can represent positive powers of ten in integer form

• The student can identify and describe patterns between powers of ten and the fraction and decimal equivalents.
  

Purposeful Questions:

• What is the pattern you recognize between a positive power of ten and its value?

• What is the pattern you recognize between the negative power of ten and its decimal value?

• How can the pattern shown help you answer the other power of ten questions?

Understanding the Learning Trajectory

Big Ideas:

• Students develop the understanding that scientific notation should be used whenever the situation calls for use of very large or very small numbers.

• A number written in scientific notation is the product of two factors — a decimal greater than or equal to 1 but less than 10, and a power of 10.

• Smaller numbers always lie to the left of larger numbers on the number line

Math Strength Instructional Video 7.1b↗

Important Assessment Look Fors:

• The student correctly compared numbers written in scientific notation that had the same exponents.

• The student correctly compared numbers written in scientific notation that had negative exponents.

• The student was able to justify their reasoning.
  

Purposeful Questions:

• How are numbers written in scientific notation with positive powers of ten different than those with negative powers of ten?

• How did you know which was the smallest number?

• How did you know which was the largest number?

• Are powers of ten with a negative exponent negative or positive numbers? Explain how you know.

Understanding the Learning Trajectory

Big Ideas:

• Rational numbers may be expressed as positive and negative fractions or mixed numbers, positive and negative decimals, integers and percents.

• Equivalent relationships among fractions, decimals, and percents may be determined by using concrete materials and pictorial representations.

• 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.

• Negative numbers lie to the left of zero and positive numbers lie to the right of zero on a number line. Smaller numbers always lie to the left of larger numbers on the number line. Rational numbers can be compared using greater than, less than, or equal.

Important Assessment Look Fors:

• The student is able to compare rational numbers using greater than, less than, or equal to symbols.

• The student is able to compare negative rational numbers.

• The student can use a model to help show the relationship between equivalent rational numbers.

• The student can determine the relationship between rational numbers and their location on the number line.

Purposeful Questions:

• How is it possible to compare/order numbers that are represented in different formats?

• How can you determine whether a number is equivalent to, greater than, or less than another number?

• How can understanding benchmark numbers, such as ½, help when comparing rational numbers?

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

Understanding the Learning Trajectory

Big Ideas:

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

• A square root of a number is a number which, when multiplied by itself, produces the given number.

• The square root of a number can be represented geometrically as the length of a side of a square.

Important Assessment Look Fors:

• Students correctly determined the values that are perfect squares.

• Students correctly determined the values that are not perfect squares.

• Students justified their reasoning for determining whether the values were perfect squares or not.

• Students justify their reasoning using connections to the area of a geometric square and its square roots.
  

Purposeful Questions:

• How did you determine the missing value for the perfect square or square root?

• Can you justify using a picture?

• How would you justify your answers using words?

Understanding the Learning Trajectory

Big Ideas:

• Numbers can be described as distance from 0.

Important Assessment Look Fors:

• The student correctly identified the absolute value of a value on the number line

• The student correctly determined the distance between two numbers on the number line and properly justified their answer.

• The student demonstrates the understanding of absolute values by correctly identifying positive and negative values with the same absolute value.

Purposeful Questions:

• Why is the absolute value of a rational number always a positive?

• How can you verify your answer on the number line?

• How does absolute value apply to finding the difference between two numbers?

Understanding the Learning Trajectory

Big Ideas:

• Just as it is important to expose students to a variety of problem situations with whole numbers, it is important to expose students to a variety of problem situations with rational numbers. To develop students’ problem solving abilities with rational numbers, use familiar contextual problems like using fractions with baking or measuring fabric, using percent for discounts or tips, and using decimals for budgeting money (Cramer & Whitney, 2010; Lewis, Gibbons, Kazemi, & Lind, 2015).

• Estimations can help students keep the focus on the meanings of rational numbers and the operation. Benchmarks can help mentally add or subtract rational numbers.

• Relate problems like multiplying a whole number by a fraction(5 x ½) with equal group problems where one iterates the fractional part. Then relate problems multiplying a fraction by a whole number (½ of 12) as partitioning or as an operator (Lamon, 2012). Relate the different meanings for operation like partitive and measurement models for division when modeling division problems with fractions.

Important Assessment Look Fors:

• The student displayed evidence of using a problem-solving strategy.

• The student understood the necessary operation(s) to solve the problem.

• The student’s implementation of their chosen strategy resulted in the correct solution.

• The student justified their answer through mathematical reasoning or visual representation.

Purposeful Questions:

• How do you know which operation is needed given the information in the problem?

• How can you use estimation to gauge if your answer (i.e., sum, difference, product or quotient) is reasonable?

• What happens to products when multiplying by a fraction? What happens to quotients when dividing by a fraction?

• What situations have you experienced where you compute fractions, decimals and/or percent in your everyday life?

Understanding the Learning Trajectory

Big Ideas:

• A proportion is a statement of equality between two ratios.

• Equivalent ratios arise by multiplying each value in a ratio by the same constant value.

• Ratio tables are an effective way to show proportionality.

• In a proportional relationship, two quantities increase multiplicatively. One quantity is a constant multiple of the other.

• A proportion is an equation which states that two ratios are equal. When solving a proportion, the ratios may first be written as fractions.

Important Assessment Look Fors:

• The student created a correct proportion using the given ratio and the additional value.

• The student is able to determine if there is only one or multiple steps to finding a solution.

• The student showed a direct relationship between the ratios and their answer.

• The student justified their response in writing/proportions/ratio table/picture.

• The student was able to estimate to determine reasonableness of the solution.

Purposeful Questions:

• How did you determine the length/width of the actual object? Of the object in the scale drawing?

• How does the scale help you determine the actual length/width of the object?

• What is the direct relationship between the picture and the actual object’s length?

• What is the factor that you are multiplying by in the task?

• What is another way to solve this proportion? How can you check your solution?

Understanding the Learning Trajectory

Big Ideas:

• The volume of a three-dimensional figure is a measure of capacity and is measured in cubic units. The development of the volume of a rectangular prism formula is parallel to the development of the area of a rectangle. The area of the base determines how many cubes can cover the base, forming a single unit, a layer of cubes. The height of the box then determines how many of the layers will fit in the box (Van de walle et al., 2019)

• The volume of rectangular prisms and cylinders are analogous in that both are multiplied by the height but different because of the type of shape on the base. Students should be encouraged to describe and connect the volume and surface area using concrete objects, diagrams, and formulas.

• To determine the surface area of rectangular prisms and cylinders, one can use concrete objects, nets, diagrams, and formulas. One can draw a net and calculate the area of each face then add up the area of all the faces.

• According to the Volume Learning Trajectory-a “3-D Row and Column Structurer” is a) Able to coordinate flexibly filling, packing, building aspects of volume. Shows a propensity for additive comparisons (e.g., “this one has 12 more”) but may show some nascent multiplicative comparisons (e.g., “this one is four times as big”); b) Initially counts or computes (e.g., number of rows times number of columns) the number of cubes in one layer (1xmxn), and then uses addition or skip counting by layers to determine the total volume. Eventually moves to multiplication (e.g., number of cubes in a layer times number of layers); c) Operates fluidly and flexibly on units (cubes), units of units (rows or columns), and units of units of units (layers). Composes and decomposes array ←→ layers ←→ rows/columns ←→ units. With perceptual support, can decompose 3-D arrays into other, complex 3-D arrays (not only layers, rows, or columns) and calculate the number of these smaller arrays in the larger array. (See more on this LT -at https://www.childrensmeasurement.org/volume.html

Important Assessment Look Fors:

• The student is able to utilize the properties of a rectangle to determine the side lengths where the amount isn’t given.

• The student uses the correct formula to determine the area of the given shape.

• The student is able to compute the surface area by either adding each two-dimensional component, or by using the distributive property to account for the opposing faces of the shape, and then adding sums of the opposing faces.

• The student uses three dimensions, l x w x h, to find the volume of the prism or finds the area (A) of the base then multiplies the height, A x h = Volume.

Purposeful Questions:

• How do you know the measurement for each side length?

• How many smaller shapes did you visualize separating the rectangular prism into to find your answer?

• Why might your answer be in a different dimensional unit when finding an area versus a surface area?

• How can you use the area of the base to help you find the volume?

• How is the volume of the prism and the cylinder similar or different?

• One friend says the surface area for a cylinder is A=2πrh+2πr² and another friend says, it is 2πr(h + r). Who is correct? How can you prove your answer?

Understanding the Learning Trajectory

Big Ideas:

• 3D solids can often be described using a 2D net and measurements from 2D shapes.

• Measurements (such as length, circumference, diameter, etc.) can be used to describe 3D solids and to determine the surface area and volume of 3D solids.

Important Assessment Look Fors:

• The student correctly identified the object and formula needed for a given situation

• The student correctly applied the formula for surface area or volume and solved accordingly

• The student correctly identified the dimensions of a rectangular prism or cylinder and substituted the values in for the appropriate variables within the surface area or volume formulas.

Purposeful Questions:

• What words helped you identify the formula needed for this problem?

• If you labeled your given values differently, how will your answer change? Can you label your given values differently without changing the solution?

• How can you justify your solution?

Understanding the Learning Trajectory

Big Ideas:

• Similar polygons have corresponding sides that are proportional and corresponding interior angles that are congruent.

• Similarity has practical applications in a variety of areas, including art, architecture, and the sciences.

• Similarity does not depend on the position or orientation of the figures.

• Congruent polygons have the same size and shape. Corresponding angles and sides are congruent.

Important Assessment Look Fors:

• The student identified corresponding sides and corresponding congruent angles of similar quadrilaterals and triangles.

• The student was able to write similarity statements given two similar quadrilaterals or triangles.

• The student wrote proportions to express the relationships between the lengths of corresponding sides of similar quadrilaterals and triangles.

• The student solved a proportion to determine a missing side length of similar quadrilaterals or triangles.

• The student determined the unknown angle measures in a similar quadrilateral or triangle.

Purposeful Questions:

• How did you determine which angles correspond with angles M and R?

• Does the orientation of the shapes matter when identifying corresponding parts?

• How can understanding the concept of congruence help with this task?

• Did you change the orientation of the figures to find the missing angle measures?

• How do you know your answer is reasonable? Justify your thinking.

Understanding the Learning Trajectory

Big Ideas:

• Classification categorization, sets and non-sets are used to group things in our world.

Important Assessment Look Fors:

• The student can correctly determine which properties apply to each type of quadrilateral

• The student can determine similarities and differences between two or more quadrilaterals based on the quadrilaterals’s properties

• The student can determine lines of symmetry for each type of quadrilateral

Purposeful Questions:

• Which shapes share properties with parallelograms?

• How do you determine whether a shape is a square or a rhombus?

• How are these two types of quadrilaterals similar? How are they different?

Understanding the Learning Trajectory

Big Ideas:

• Quadrilaterals are classified and categorized based on properties such as equivalent or parallel.

• Properties can be used to determine unknown quantities.

Important Assessment Look Fors:

• The student can apply the properties of the sides of quadrilateral to practical problems

• The students can apply the properties of the angles of quadrilateral to practical problems

Purposeful Questions:

• Which property of quadrilaterals did you apply to solve this problem?

• What information in the problem helped you determine the quadrilateral that applied to this question?

Understanding the Learning Trajectory

Big Ideas:

• A transformation of a figure called the preimage changes the size, shape, or position of the figure to a new figure called the image.

• Translations and reflections do not change the size or shape of a figure (e.g., the preimage and image are congruent figures).

• Translations and reflections change the position of a figure. A translation is a transformation in which an image is formed by moving every point on the preimage the same distance in the same direction.

• A reflection is a transformation in which an image is formed by reflecting the preimage over a line called the line of reflection. All corresponding points in the image and preimage are equidistant from the line of reflection.

Important Assessment Look Fors:

• The student can identify the coordinates of the image of a right triangle or rectangle that has been translated or reflected when given a preimage.

• The student can sketch the image of a right triangle or rectangle that has been translated vertically, horizontally, or a combination of both.

• The student can sketch the image of a right triangle or rectangle that has been reflected over the x- or y-axis.

• The student can sketch the image of a right triangle or rectangle that has been translated and reflected over the x- or y-axis or reflected over the x- or y-axis and then translated.

Purposeful Questions:

• What is different between the coordinates of the preimage and new image when you reflect an image across the X-axis?

• When you translate a point to the right 5 units, what happens to the ordered pair?

• When you translate a point to down 4 units, what happens to the ordered pair?

• What do you notice about the coordinates of a point when it is reflected over the x-axis?

• What do you notice about the coordinates of a point when it is reflected across the y-axis?

Understanding the Learning Trajectory

Big Ideas:

• CHANCE: 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.

• Probability can provide a basis for making predictions. Some probabilities can only be determined through experimental trials. An event that is certain to happen will always happen (the probability is 1.) and an event that is impossible will never happen (the probability is 0.)

• The theoretical probability is what SHOULD happen and experimental probability is the actual RESULT (outcome) of a trial and probability of an event can be represented as a ratio or equivalent fraction, decimal, or percent.

Important Assessment Look Fors:

• The student can determine the theoretical probability of an event.

• The student can determine the experimental probability of an event.

• The student can show that probability can be expressed as fractions/decimals/percents.

• The student shows understanding that frequency directly impacts experimental probability.

Purposeful Questions:

• How do you determine a sample space in a given probability situation?

• How do you determine if an event is impossible, less likely, more likely, certain?

• How can you represent the probability of an event using a fraction/decimal/percent?

• How is the experimental probability different from the theoretical probability?

Understanding the Learning Trajectory

Big Ideas:

• CHANCE: 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.

• Probability can provide a basis for making predictions. Some probabilities can only be determined through experimental trials. An event that is certain to happen will always happen (the probability is 1.) and an event that is impossible will never happen (the probability is 0.)

• The theoretical probability is what SHOULD happen and experimental probability is the actual RESULT (outcome) of a trial and probability of an event can be represented as a ratio or equivalent fraction, decimal, or percent.

• In experimental probability, as the number of trials increases, the experimental probability gets closer to the theoretical probability (Law of Large Numbers).(VDOE, Curriculum Framework)

Important Assessment Look Fors:

• The student can determine the difference between theoretical and experimental probability given an event.

• The student can describe changes in the experimental probability as the number of trials increases.

• The student can investigate and describe the difference between the probability of an event found through experiment or simulation versus the theoretical probability of that same event.

Purposeful Questions:

• What is the theoretical probability for this event to occur?

• What is the experimental probability for this event to occur?

• What do you think would happen to the experimental probability if you increased your number of trials? How would it compare to the theoretical probability?

• How is experimental probability different from theoretical probability? Provide an example.

Understanding the Learning Trajectory

Big Ideas:

• Histograms use bin size to display data in meaningful ways.

• Histograms, similar to bar graphs, use rectangular bars whose heights correspond to frequency. However, bar graphs plot categorical data and have gap between each bar, whereas histograms plot numerical data and are continuous (no gaps).

• The student can represent data through frequency intervals in a histogram

Important Assessment Look Fors:

• (7.9a)-The student can display data in a histogram given data in a table format or list format

• (7.9a) The student can display data frequency accurately with given intervals in a histogram

• (7.9c) The student can make conclusions regarding information that can be inferred from different graphical representations of the same data.

Purposeful Questions:

• How did you decide to use this bin size?

• How does the stem and leaf plot look similar and different to the histogram?

• What information can the stem-and-leaf graph [circle graph, line plot] provide you that the histogram cannot?

Understanding the Learning Trajectory

Big Ideas:

• A histogram is used to deduce information and analyze data when presented in consecutive intervals.

• Comparisons, predictions and inferences are made by examining characteristics of a data set displayed in a variety of graphical representations to draw conclusions.

• Data analysis helps describe data, recognize patterns or trends, and make predictions.

• Inferential statistics uses data in a sample selected from a population to describe features of the population (Kader & Jacobbe, 2013. NCTM, Essential Understandings Statistics 6-8).

• The sampling distribution of a statistic describes the sample-to-sample variability and values of the statistic from the multiple samples of the same size selected from the same population.

• Selecting a simple random sample from a population is a fair way to select a sample.

• The predictable pattern for the sampling distribution of a statistic based on random sampling provides a way for making inferences about a population.

Important Assessment Look Fors:

• The student shows understanding of the verbiage of “no fewer than” means “equal to or more.”

• The student recognizes that there is more than one interval that serves a solution.

• The student computes the combination of the occurrences in each appropriate interval, and not the amount represented by the interval.

• The student is able to create a proper representation of the number of successful solutions to the scenario in comparison to the total number of cities represented.

Purposeful Questions:

• Do you know all of the exact data values? Why or why not?

• Which measures of center could be computed from a histogram? Explain your reasoning.

• Which intervals would be included in fewer than___? At least_____? Greater than ____? How do you know?

• How will you convert the data from the histogram to a percentage?

Understanding the Learning Trajectory

Big Ideas:

• When two quantities, x and y, vary in such a way that one of them is a constant multiple of the other, the two quantities are “proportional.”

• The slope of a proportional relationship can be determined by finding the unit rate.

• The slope of a line is a rate of change, a ratio describing the vertical change to horizontal change of the line.

• Linear equations can be solved by symbolic, graphical, and numerical methods. On some occasions and in some contexts, one solution method may be more elegant, efficient, or informative than another (Lloyd, Herbel Eisenmann, Star & Zbiek, 2011, NCTM’s Essential Understanding of Expressions, Equations and Functions 6-8)

Important Assessment Look Fors:

• The student is able to determine the slope of a proportional relationship by finding the unit rate.

• The student is able to determine the slope of proportional relationship by using a ratio that describes the vertical change to horizontal change of the line.

• The student is able to describe a proportional relationship from a table.

• The student is able to describe a proportional relationship from a verbal description.

Purposeful Questions:

• How could you represent a proportional relationship?

• How did you determine the slope from the table?

• How did you determine the equation from the table?

• How can you represent this problem situation using algebraic symbols, numbers, graphs, verbal descriptions, and tables?

Understanding the Learning Trajectory

Big Ideas:

• When two quantities, x and y, vary in such a way that one of them is a constant multiple of the other, the two quantities are “proportional.”

• The slope of a proportional relationship can be determined by finding the unit rate.

• The slope of a line is a rate of change, a ratio describing the vertical change to horizontal change of the line.

Important Assessment Look Fors:

• The student utilizes the origin and the given slope to locate another point in a given graph.

• The student is able to insert a chosen input value into the equation of the line, and determine an output value.

• The student can consistently determine the x-value given the y-value, or find the y-value given the x-value in a linear equation.

Purposeful Questions:

• How will knowing the slope of the line help you to figure out an initial set of points?

• What strategy do you prefer to use when selecting three other points on the line? Explain your strategy.

• Is the incline of the line between all of your points identical? Why is this important?

Understanding the Learning Trajectory

Big Ideas:

• The y-intercept of a line represents the initial value in a relationship between two variables.

• Two quantities, x and y, have an additive relationship when a constant value, b, exists where y = x + b, where b does not equal 0.

• The y-intercept of an additive relationship can be determined by adding a value to the other quantity.

Important Assessment Look Fors:

• The student is able to determine the y-intercept of an additive relationship from a table, graph, and equation.

• The student is able to write the equation of a line of an additive relationship from a table, graph, and equation.

• The student is able to identify the y-intercept of an additive relationship by adding a value to the other quantity in a table.

Purposeful Questions:

• How could you represent an additive relationship?

• What is the difference between an additive relationship and a proportional relationship?

• How did you determine the y-intercept from the table?

• How did you determine the equation of the line?