Understanding the Learning Progression

Big Ideas:

• The base-ten system helps students see a relationship between adjacent place values which in turn helps them compare decimals and thus supports their ability to round them. For example, it is important to deepen understanding and fluency with decimals in the different forms, seeing .57 as 5 tenths and 7 hundredths as well as 57 hundredths (Common Core Standards Writing Team, 2019, p. 64). This ability to rename and decompose decimals can help students round to the nearest whole number, tenth or hundredth.

• A decimal point separates the whole number and decimal places. Place values extend infinitely in two directions from a decimal point.

• In mathematics, decimals can be written correctly by remembering that any decimal less than one can include a leading zero (e.g., 0.125). This number may be read as “zero and one hundred twenty-five thousandths” or as “one hundred twenty-five thousandths.”

• A decimal number lies between other decimal places and/or whole numbers. For example 5.65 lies between 5.6 and 5.7 as well as whole numbers 5 and 6.

Important Assessment Look-fors:

• The student writes the decimal quantity accurately, placing the decimal point correctly.

• The student uses base-10 models to support their reasoning for rounding.

• The student uses a number line to round a decimal. The student locates a number on the number line, determines the closest multiples of whole numbers, tenths, or hundredths that it lies in between, and identifies which it is closer to. In other words, the student can determine the consecutive whole numbers/tenths/hundredths between which a given number lies.

• The student determines numbers that round or do not round to a given benchmark.

Purposeful questions:

• How did you determine the start, end, and midpoint of your number line?

• How did you determine the relative location of your decimal?

• How did you decide which location to round to on your number line? Why did you decide to round in that direction?

• What do you look at when rounding to the nearest tenth? To the nearest hundredth? To the nearest whole number?

• How would your answer change if you rounded to a different place value?

Understanding the Learning Progression

Big Ideas:

• 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).

• Any number can be represented in an infinite number of ways that have the same value (Charles, p.10).

• Decimal to fraction equivalents can be named not just through procedures but through sense-making with our base-ten system (e.g. ⅖ = 4/10 = 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 equivalent (e.g. 12/50=24/100=0.24)

• Naming an equivalent fraction and decimal means that the quantities are the same even though they are represented differently. ¾ is 0.75 written in a different form. Both ¾ 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.

Important Assessment Look-fors:

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

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

• The student demonstrates an understanding that fractional models can also be written in many equivalent decimal forms (0.8, 0.80, etc.) and vice versa.

• The student recognizes that there is more than one way to name an equivalent fraction or decimal to represent a model or quantity including in simplest form.

Purposeful questions:

• What strategies did you use to find a decimal equivalent for common fractions (e.g., 1/2 , 1/4, 1/8,2/4, 3/4 or 1/5, 1/10) ? Which strategy is the most efficient for you and why? How can one connect fraction and decimal with money, like quarter, two quarters, three quarters?

• How can you connect fractions and decimals with money, like quarter, two quarters, three quarters?

• What strategies did you use to find a fraction equivalent for decimals? Which strategy is the most efficient for you and why?

• How does a decimal grid, fraction bars, or rational number wheel help you find a fraction or decimal equivalent?

• How can finding a decimal fraction with the denominators as 10, 100 or 1000 help you change fractions to decimals (e.g.,1/4 =25/100=0.25 and 1/8=125/1000=0.125)?

• How might division help you find a decimal equivalent for a fraction?

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 (e.g., 5/9 compared to 6/9) and also when the denominator increases (e.g., 2/4 compared to 2/5). 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 and decimals are essentially the same numbers in different forms, they can be compared and ordered. Fractions and decimals can be compared and ordered using a variety of strategies including using benchmarks (0, halves, wholes), drawing representations, placing them on a number line, naming equivalencies, and other reasoning strategies.

• Decimal to fraction equivalents can be named not just through procedures but through sense-making with our base-ten system (e.g. ⅖ = 4/10 = 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 (e.g. 12/50=24/100=0.24).

Important Assessment Look-fors:

• The student uses an efficient strategy/strategies to order a set of decimals or fractions in seclusion (e.g. 5.009, 5.6, 5.67, 5.75).

• The student uses multiple strategies to compare and order fractions and decimals (benchmarks, equivalencies, close to a whole).

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

• The student uses mathematical symbols <, >, = or ≠.

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?

• For which problems is it most efficient to use benchmarks to compare and order and for which did you find it necessary to do renaming? Why?

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

Understanding the Learning Progression

Big Ideas:

• Prime numbers are natural numbers that have exactly two different factors: 1 and itself. Composite numbers are natural numbers that have three or more different factors.

• Every composite number can be expressed as the product of prime numbers in exactly one way, disregarding the order of the factors (Fundamental Theorem of Arithmetic; Charles, 2005). Prime numbers are thus the building blocks of all composite numbers.

• The number 1 is a unique natural number in that it is neither prime nor composite due to it only having 1 factor (1x1=1).

• Looking for patterning in multiplication facts and relating these to divisibility rules can help students efficiently determine if a number is prime or composite.

Important Assessment Look-fors:

• The student accurately draws a model that represents that a number is prime or composite and explains their thinking. This could include an array, a list of factors, or equations .

• The student recognizes and explains general patterns within prime or composite numbers, but points out where these patterns have exceptions. For example, all even numbers are composite except for 2 or while many prime numbers are odd not all odd numbers are prime.

• The student recognizes that 1 is neither prime nor composite because it only has 1 factor.

Purposeful questions:

• How can one prove that a number is prime or composite?

• What strategy do you use to determine all of the factors of a given number?

• How are prime numbers similar or different from composite numbers?

Understanding the Learning Progression

Big Ideas:

• PATTERNS: Relationships can be described and generalizations made for mathematical situations that have numbers or objects that repeat in predictable ways. Known elements in a pattern can be used to predict other elements (Charles, 2005).

• All whole numbers can be categorized into the subset of odd or even numbers. These numbers follow an infinite pattern of even, odd.

• All whole numbers are divisible by 2. A number is even only if it can be divided by 2 and results in a whole number answer. (For example, 7 is not an even number although it can be broken into two equal groups of 3.5).

• Even numbers are divisible by 2 and thus a rule can be created that all even numbers have a 0, 2, 4, 6, or 8 in the ones place; while all odd numbers are not divisible by 2 and therefore have a 1, 3, 5, 7, or 9 in the ones place.

• When finding the sum of two whole numbers, one can predict whether the answer will be odd or even. The sum of two even numbers or two odd numbers will always result in an even number. The sum of an odd and even number will always result in an odd number.

Important Assessment Look-fors:

• The student determines if a series of two digit numbers written in standard form are odd or even.

• The student draws a model to represent a number as even by being able to be broken into groups of 2 or 2 equal groups (demonstrate divisibility by 2), rather than simply relying on rules such as “Numbers with a 0, 2, 4, 6, or 8 in the ones place are even.”

• The student uses a model to represent how the sum of two odds will always result in an even number. The student should be able to represent the “leftovers” of an odd number coming together with another and explain why this will always happen.

• The student draws an accurate depiction of the difference between an odd and even number and supports their model with a description or equation(s).

Purposeful questions:

• What is an efficient strategy for determining if a number is odd or even? Can you do this without having to draw a picture?

• How can you use a model to prove your answer?

• Can you create multiple equations that will prove or disprove your thinking?

Understanding the Learning Progression

Big Ideas:

• By reasoning repeatedly about the connection between math drawings and written numerical work, students can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning (Common Core Writing Team, 2019, p. 14)

• The relationship between inverse operations allows students to make decisions about which operations to use to solve problems. Understanding those relationships will support students’ reasoning about problem solving.

• The context of a problem determines the meaning of a remainder and how the remainder affects the solution to the problem.

• 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.

• When solving problems, using unit labels with drawings, symbols, numbers will support students’ decision making and reasoning about appropriate solutions.

Important Assessment Look-fors:

• The student determines an appropriate operation to use in a single-step word problem.

• The student uses pictures, numbers, or words to represent and explain the process to solve the problem.

• The student determines the operations of a multi-step word problem and chooses an appropriate plan of action to solve.

• The student labels the units throughout the problem and in the answer to determine reasonableness (in division problems with and without remainders).

Purposeful questions:

• What are the units? What is being counted in the problem?

• Is the total known or unknown?

• 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 your answer mean?

• What do the groups represent? How many are in each group?

Understanding the Learning Trajectory

.

Big Ideas:

• Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones. (Charles, 2005). For example, when dividing a decimal by a decimal, you can multiply both the dividend and divisor by the same powers of ten to work with whole numbers.

• Numerical calculations can be approximated by replacing numbers with other numbers that are close and easy to compute with mentally (Charles, 2005). This estimation can be used to determine a reasonable range for the answer and to verify it’s reasonableness.

• Division is the operation of making equal groups or shares. The fair-share concept of decimal division can be modeled, using manipulatives (e.g., base-ten blocks), arrays, paper folding, repeated addition, repeated subtraction, base-ten models, and area models.

• Algorithms for whole number multiplication and division can be used to help make sense of decimal number multiplication and division.

Important Assessment Look Fors:

• The student uses estimation and rounding in order to determine where to place the decimal in the product or quotient.

• The student models multiplication and division of decimals with various models as well as through computation. The student may champion a particular efficient strategy.

• The student interprets a model to determine what the product or quotient it is representing.

Purposeful Questions:

• How did you determine where the decimal should be placed in your product/quotient? How do you know this is a reasonable answer?

• When solving division problems, why do numbers need to be expressed as equivalent decimals by annexing zeros?

• How does this model represent the situation? Where do you see the (factors, product, dividend, divisor, quotient) represented?

• How is estimating with decimals similar or different to estimating with whole numbers?

Understanding the Learning Trajectory

Big Ideas:

• Numerical calculations can be approximated by replacing numbers with other numbers that are close and easy to compute with mentally (Charles, 2005). This estimation can be used to determine a reasonable range for the answer and to verify its reasonableness.

• There are a variety of algorithms that can be used for the four processes. Students should utilize the strategy that works best for them with whole numbers and determine how to use it appropriately to solve expressions with decimal numbers.

• Multiplication and division have an inverse relationship. This relationship can become confusing when students start to multiply and divide with decimals. Multiplying a whole number times a decimal less than 1 results in a product smaller than the number being multiplied because we are finding a fractional amount of a quantity. When dividing a number by a decimal less than 1, the quotient is greater. Students need opportunities to use manipulatives to make sense of why this is and how it works.

Important Assessment Look Fors:

• The student checks their work for reasonableness based on estimating and rounding.

• The student uses efficient strategies to add, multiply, subtract, and divide with decimal numbers.

• The student interprets a multistep problem and determines a plan of action.

Purposeful Questions:

• When you started adding and subtracting with decimal numbers, what did you do with the decimal point? Why?

• Did you do some estimation in order to determine where to start to solve the problem? Why or why not? How could estimates have helped you?

• How is adding or subtracting with whole numbers similar or different to adding and subtracting with decimal numbers?

• How is multiplying or dividing with whole numbers similar or different to multiplying and dividing with decimal numbers?

Understanding the Learning Trajectory

Big Ideas:

• To name equivalent fractions students see that multiplying the numerator and denominator of a fraction by the same number, n, corresponds to partitioning each piece of the diagram into n smaller equal pieces. (Arizona, 10)

• Converting an improper fraction to a mixed number is a matter of decomposing the fraction into a sum of a whole number and a number less than 1 (Common Core Writing Team, 2019, p. 12).

• When adding and subtracting fractions, regrouping is necessary based on the “whole” unit. For example, when finding the difference of 6 - 4 ¾ you must regroup 1 whole for an equivalent mixed number of 5 and 4/4. The same can be said when regrouping in addition.

• The strategies students utilize to break apart and make sense of word problems with whole numbers can be utilized to make sense of word problems with fractions.

• 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).

Important Assessment Look Fors:

• The student can name equivalent fractions by multiplying the numerator and denominator by the same factor (or 1).

• The student can rename an improper fraction as a mixed number and vice-versa.

• The student can reason about regrouping with fractions as renaming a whole into an equivalent whole made of n number of pieces (1 whole = n/n).

• The student can correctly determine the operation of a single-step word problem.

• The students can use pictures, numbers, or words to represent and explain the process to solve the problem.

• The student can correctly determine the operations of a multi-step word problem and determine a plan of action to solve.

• The student can estimate to check their answer for reasonableness.

Purposeful Questions:

• How does renaming that fraction as an equivalent help you? What are you really doing when you rename it?

• How did you use the manipulatives/drawings/paper etc. and come up with a strategy for solving? How can you write an equation to match your drawing?

• How is regrouping with fractions similar or different to regrouping with whole numbers?

• Is the total known or unknown?

• 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 your answer mean?

Understanding the Learning Trajectory

Big Ideas:

• Previously students have seen that 3 x 7 can be represented as the number of objects in 3 groups of 7 objects, and write this as 7 + 7 + 7. Students apply this understanding to fractions, seeing

• 1/3 + 1/3+ 1/3+ 1/3 + 1/3 as 5 x 1/3. This allows students to give meaning to the product of a whole number and a fraction (Common Core Writing Team, 2019, p. 14).

• All fractions are a sum of their unit fractions. For example, ¾ = ¼ + ¼ + ¼ .

• Fraction operation should begin with Multiplying a whole number by a fraction-specifically a whole number by unit fractions (e.g., 3 x 1/3) then move to multiplication by whole number by nonunit fractions (e.g., 3 x 2/3)

• Multiplying unit fraction by a whole number can be related to dividing the whole number by the denominator of the fraction. For example, 13of 6 is equivalent to 2. This understanding forms a foundation for learning how to multiply a whole number by a proper fraction (5th grade Curriculum Framework, p. 21).

• Multiplying a whole number times its reciprocal will result in a product of one whole. (Example 5 x 1/5 = 5/5 or 1; 1/6 x 6 = 6/6 or 1).

Important Assessment Look Fors:

• The student can write repeated addition of a fraction as the product of a fraction and a whole number.

• The student can create concrete and pictorial models to represent and simplify an expression.

• The student can interpret concrete and pictorial models to represent and solve an expression.

• The student can use the model to justify why the solution makes sense.

• The student can simplify their answer.

Purposeful Questions:

• What do you notice about the product of a fraction and a whole number? Why? (Students should notice it is smaller than the original whole because they are taking only PART of that whole).

• How does your picture represent repeated addition? Where do you see multiplication?

• Where do you see division in your model? How does this relate to the expression/equation you wrote? How does this represent the story in the problem?

Understanding the Learning Trajectory

Big Ideas:

• EQUIVALENCE: Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value. (Charles, 14)

• The order of operations is a convention that defines the computation order to follow in simplifying an expression. It ensures that there is only one correct value. If we did not have an order of operations everyone would get different solutions to the problem.

• The order of operations gives structure and order to a situation. Utilization of this process helps us represent multiple expressions in one complex expression.

• The order of operations utilizes a variety of notations in order to represent operations.

Math Strength Instructional Video↗

Important Assessment Look Fors:

• The student explains and reasons that inverse operations (addition/subtraction; multiplication/division) have equal importance when simplifying an expression.

• The student describes which operation is completed first, second, etc., and why in a given whole number-based numerical expression involving more than one operation.

• The student explains that the order of operations is a convention used so that all mathematicians can come to the same conclusion when solving an expression.

• The student shows work step-by-step in a logical manner to be able to check work for accuracy.

Purposeful Questions:

• Why is the Order of Operations a convention that must be used when solving an expression?

• What common errors do you think students make when solving an expression such as this? What hint or “look-fors” would you give them before they solved it?

• How are you keeping track of which operations/steps you have completed? How does this help you?

Understanding the Learning Trajectory

Big Ideas:

• Area of a shape (in square units) is the number of unit squares it takes to cover the shape without gaps or overlaps (Common Core Standards Writing Team, 2019).

• Perimeter, area, and volume are all measurements of space in either 1, 2, or 3 dimensions.

• The area of a right triangle is always half that of the area of a rectangle with the same base and height.

• The formula for the volume of a rectangular prism can be discovered by reiterating the area of one layer of the rectangular prism repeatedly over itself. Therefore the volume of a rectangular prism is the area of one layer of that prism multiplied by the height of the prism.

Important Assessment Look Fors:

• The student differentiates between perimeter, area, and volume.

• The student recognizes the relationship between triangles and rectangles.

• The student uses an efficient strategy to find the perimeter (add all sides or 2L + 2W), area (L x W) and volume.

• The student determines a way of solving each problem using the information given as well as manipulatives in their classroom.

Purposeful Questions:

• How did you determine which lengths/dimensions/numbers you needed to use to find your answer? (Ex: Square, I only see 1 number, how did you know to use the number multiple times?)

• How did you determine what operation to use and why?

• How does the picture you drew support your strategy to solve the problem?

• Why (or when) can you use multiplication to find the perimeter, area, and volume of a rectangle/rectangular prism?

• How is a triangle related to a rectangle in terms of its area?

Understanding the Learning Trajectory

Big Ideas:

*Note: This standard focuses on volume as it relates to BOTH the volume of a rectangular prism and liquid volume since the standard focuses on real-life application and not on solving for the volume.*

• Area of a shape (in square units) is the number of unit squares it takes to cover the shape without gaps or overlaps (Common Core Standards Writing Team, 2019, p. 17).

• Perimeter, area, and volume are all measurements of space in either 1, 2, or 3 dimensions. Real-world application requires students analyzing what measurement of space they are trying to find based on the unique situation, using the information given to determine a process for solving, and then applying the correct units to their solution.

• Perimeter, area, and volume all use similar information to determine their solutions (length, width, and/or height). Students must be able to determine which information is important and/or superfluous for solving the given situation.

• The formula for the volume of a rectangular prism can be discovered by reiterating the area of one layer of the rectangular prism repeatedly over itself. Therefore the volume of a rectangular prism is the area of one layer of that prism multiplied by the height of the prism.

Important Assessment Look Fors:

• The student describes a practical situation where perimeter, area, and volume are appropriate measures to use and justifies their answer.

• The student uses pictures, numbers, and/or words to show the relationship between perimeter, area, and volume and uses the dimensions (length, width, and height) to show relationship.

Purposeful Questions:

• What helped you determine if this scenario was asking about (perimeter, area, volume)?

• Which dimensions of the figure/item you are using in your example will be used to find the perimeter, area, and/or volume?

• What common misconceptions do you think someone who is new to this learning may have?

Understanding the Learning Trajectory

Big Ideas:

Note: This standard 5.9b focuses on students’ ability to estimate and measure to solve practical problems that involve metric units, while 5.9a focuses on naming equivalencies.

• The metric system is a logical base-ten system of measurement in which the base measurements are the meter (length), liter (volume), and gram (mass/weight).

• Naming an equivalent measure involves taking a unit and multiplying or dividing by powers of 10.

• Estimating measurement equivalencies is necessary in order to check the accuracy of one’s measurements.

• When measuring an object's attributes (length, volume, weight) it is necessary to choose an appropriate unit and tool for the job. For example when measuring the length of a car one must determine the appropriate unit (meters) as well as the appropriate tool (meter stick, tape measure).

• “There are prefixes for multiples of the basic unit (meter or gram), although only a few (kilo-, centi-, and milli-) are in common use,” (Common Core Standards Writing Team, 2019, p. 20).

• Input-output tables can be used to show relationships between numbers. In the metric system, input-output charts can be used to show the base-ten (or powers of ten relationship) that exists between units. Students need experience analyzing a table to determine the rule, name the input when given the output, and vice-versa.

Important Assessment Look Fors:

• The student uses a given equivalency (e.g. 10 millimeters = 1 centimeter) to provide reasoning the equivalent amount for any number of centimeters.

• The student completes an input-output table when given the equivalency (or rule) needed. Note that sometimes these equivalencies are given in one direction (meters to kilometers) but the table is in the opposite order (kilometers to meters).

• The student completes both a vertical and horizontal input-output table whether the included data is in the x or y column.

• The student names equivalencies when the given number is whole or decimal number.

• The student applies a given rule to the input to find the output, and applies the inverse to the output the find the input.

Purposeful Questions:

• What important information do they give you that can help you with the equivalencies below?

• In what way is the information ordered?

• What pattern or relationship do you see between the input and the output? How can you use that pattern to fill out the table?

• How does the fact that the number in the input/output is a decimal number affect your strategy for solving?

Understanding the Learning Trajectory

Big Ideas:

Note: This standard focuses on students’ ability to estimate and measure to solve practical problems that involve metric units, while 5.9a focuses on naming equivalencies.

• The magnitude of the attribute to be measured and the accuracy needed determines the appropriate measurement unit (Charles, 2005). Measuring length, mass, and volume requires students to determine first which metric measurement they are trying to find, then pick an appropriate instrument to measure accurately. Finally, students must choose an appropriate unit to label their measurement.

• Weight and mass are different. Mass is the amount of matter in an object. Weight is determined by the pull of gravity on the mass of an object. The mass of an object remains the same regardless of its location. The weight of an object changes depending on the gravitational pull at its location. In everyday life, most people are actually interested in determining an object’s mass, although they use the term weight (e.g., “How much does it weigh?” versus “What is its mass?”).

• Metric measurement units are related by tens. Students must see the fractional relationship between metric units (base 10) and relate to the prefixes (milli-, cent-, kilo-) in order to help estimate more accurately.

Important Assessment Look Fors:

• The student can determine whether they are looking to find length, mass or liquid volume.

• The student can determine options for measurement units (grams/kilograms, milliliters/liters, millimeters/centimeters/meters/kilometers), choose one unit to utilize for their estimate, and justify its use for each of the scenarios.

• The student can check their estimate or actual measurement for reasonableness.

Purposeful Questions:

• How did you determine if you were finding length, mass, or liquid volume?

• What units of measurement could you use to measure this item but which one did you decide to use and why?

• When estimating, what objects do you associate with the base units? (I.e. “I think of the weight of a paperclip for one gram or a dictionary for 1 kilogram).

Understanding the Learning Trajectory

Big Ideas:

• SHAPES & SOLIDS: Two- and three-dimensional objects with or without curved surfaces can be described, classified, and analyzed by their attributes (Charles, 2005).

• 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 chord is a line segment that connects two points on the circumference of the circle.

• 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.

Important Assessment Look Fors:

• The student correctly identifies the radius, diameter, chord, and circumference in a diagram of a circle. The student must note that there can be infinite radii, diameters, and chords on a circle.

• The student explains that a diameter is a special type of chord that goes through the center of a circle, however a radius is not a chord because although it goes to the center of the circle, it does not connect two points on the circumference of the circle.

• The student describes the relationship between the diameter and radius (diameter is twice the length of radius or the radius is half the length of the diameter) and can solve for one when given the other.

• The student explains the relationship between the diameter and circumference of the circle (the circumference is about 3 diameters) and solves for one when given the other.

Purposeful Questions:

• What do you notice about the relationship between the length of the __________ and the length of the _____________? Using this information how could you use the length of one to find the magnitude of the other?

• What are the parts of a circle and can you describe what they measure?

Understanding the Learning Trajectory

Big Ideas:

• An elapsed time problem always consists of a start time, end time, or elapsed time; two of these items are given and one is unknown.

• A variety of tools can be used to solve elapsed time word problems such as an open number line, a t-chart, a demonstration clock, or even regrouping.

• In 3rd grade, students learn how to tell time, match a digital clock to an analog clock, name how minutes in a day and how many hours in a day. They also begin to investigate elapsed time in one-hour increments within a 12 hour period (within a.m. or p.m.) 4th graders extend this learning by solving practical problems related to elapsed time in hours and minutes, within any 12-hour period while 5th graders complete elapsed time in hours and minutes within 24 hours.

• “Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones. …Times in minutes and seconds can be added and subtracted where 1 minute is regrouped as 60 seconds.” (Charles, p. 16-17).

Important Assessment Look Fors:

• The student reads the time on an analog clock.

• The student reads and diagnoses a word problem to determine what information is given and what information must be solved for (start time, elapsed time, or end time).

• The student solves for the unknown (start time, elapsed time, or end time) by using an appropriate strategy such as an open number line, subtracting/adding time, or a t-chart.

Purposeful Questions:

• How did you figure out what piece of information you were trying to solve for?

• Why did you choose this tool (open number line, subtracting/adding time, t-chart) to solve this problem? How did it help you?

• When are some tools more useful than others (open number line, subtracting/adding time, t-chart)?

• What happens to the hour hand as the minute approaches the hour?

Understanding the Learning Trajectory

Big Ideas:

• MEASUREMENT: Some attributes of objects are measurable and can be quantified using units. Measurement involves a selected attribute of an object (length, area, mass, volume, capacity) and a comparison of the object being measured against a unit of the same attribute (Charles, 2005).

• An angle is a two-dimensional representation of a rotation or the space between two rays/lines.

• A number of degrees can be used to describe the size of an angle’s opening (Charles, 2005).

• All angles can be categorized based on the size of their angle: acute, right, obtuse, or straight. (Note that higher grades/levels of math will discuss zero, reflex, and full rotations.)

• Some angles have special relationships based on their position or measures (e.g., complementary angles) (Charles, 2005).

Important Assessment Look Fors:

• The student names the appropriate tool utilized to draw and measure an angle.

• The student can accurately align, extend, estimate, and measure an acute, right, obtuse and straight angle using a protractor.

• The student notes an angle measurement with the symbol for degrees (°).

• When given the measure of one angle in a pair of supplementary angles (angles that measure to 180° - or a straight line), the student uses addition or subtraction to determine the missing angle.

Purposeful Questions:

• How are the tools we use to create an angle similar or different to the tools that we use to measure an angle?

• What steps do we need to remember in order to align our protractor correctly to measure an angle?

• How can estimation help you check the reasonableness of your answer?

• What is special about a straight angle that can help you find the missing angle pictured?

Understanding the Learning Trajectory

Big Ideas:

• SHAPES & SOLIDS: Two- and three-dimensional objects with or without curved surfaces can be described, classified, and analyzed by their attributes. 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).

• A triangle is a 3-sided polygon that can be classified by either its sides (scalene, isosceles, equilateral), its angles (acute, right, or obtuse), or a combination of both.

• The sum of the angles of a triangle is 180 degrees.

• The angles within a triangle affect the length of its sides and vice-versa.