## Formative Assessment and Bridging activities

Grade 5

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Standards are considered a bridge when they: function as a bridge to which other content within the grade level/course is connected; serve as prerequisite knowledge for content to be addressed in future grade levels/courses; or possess endurance beyond a single unit of instruction within a grade level/course.**Note:** Links marked with ↗ will open in a new tab

## Standard 5.1

**S****tandard**** ****5.1 **Given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth.

**(Pull down for more)**

Understanding the Learning Progression

**Big Ideas:**

Understanding of the base-ten system to the relationship between adjacent places and how numbers compare can help support students round for decimals to thousandths. 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 and 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 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.”

One should be able to locate the relative location of a decimal on a number line between two other decimal or whole number locations.

**Important Assessment Look-fors: **

The student can write the decimal quantity accurately, placing the decimal point correctly.

The student can show knowledge of the base-ten number system by modeling and describing the ten-to-one place value relationship with or without manipulatives.

The student can use a number line to round a decimal. Students can locate a number on the number line, determine the closest multiples of whole numbers, tenths, or hundredths that it lies in between, and identify which it is closer to. In other words, students can determine the consecutive whole numbers/tenths/hundredths between which a given number lies.

**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 you answer change if you rounded to a different place value?

**Student Strengths**

**Student Strengths**

Students can round to the nearest whole number; identify decimal place values through thousandths; and round a whole number to any given place value and make generalizations about this process.

**Bridging Concepts**

Students can round decimals expressed through tenths and hundredths to the nearest whole number. Students can name the halfway point between two decimal locations with or without a number line.**Bridging Concepts**

**Standard ****5.1**

Students can be given a decimal through thousandths and round to the nearest whole number, tenth, or hundredth.**Standard**

**5.1**

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

- Make it Close↗
- Desmos 5.1 Rounding Decimals↗

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## Standard 5.2a

**S****tandard**** 5.****2a**** **Represent and identify equivalencies among fractions and decimals, with and without models.

**(Pull down for more)**

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 (such as the one above showing ⅘) can also be written in many equivalent decimal forms (0.8, 0.80, etc.).

**Purposeful ****questions:**

What strategies do 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?

What strategies do 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 one change fractions to decimals (e.g.,1/4 =25/100=0.25 and 1/8=125/1000=0.125)?

How might division help one find a decimal equivalent for a fraction?

**Student Strengths**

**Student Strengths**

Students can name equivalent fractional amounts using concrete and pictorial models less than one.

**Bridging Concepts**

Students can name fractions with denominators of 2, 4, 5, 10, 20, 25, and 50 as equivalent fractions with a denominator of 100 and record in decimal form. **Bridging Concepts**

**Standard 5.****2a**

Students can represent and identify equivalencies among fractions and decimals, with and without models.**Standard 5.**

**2a**

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **

**Rich Tasks:**

How Much Money Goes Into the Savings Jar? by MathStrength↗

**Games/Tech:**

**Games/Tech:**

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## Standard 5.2b

**S****tandard**** 5.2****b**** **Compare and order fractions, mixed numbers, and/or decimals in a given set, from least to greatest and greatest to least

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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), 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 to find the decimal equivalent (e.g. 12/50=24/100=0.24).

Math Strength Instructional Video↗

**Important Assessment Look-fors: **

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

The student can determine a fraction or decimal number 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?

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 fraction similar or different to the strategies you use to compare decimals?

**Student Strengths**

**Student Strengths**

Students can compare 2 fractions using the symbols <, >, = and compare 2 decimals using the symbols <, >, =.

**Bridging Concepts**

Students can compare and order 3 fractions or 3 decimals from least to greatest or greatest to least. Students can compare 1 fraction and 1 decimal.**Bridging Concepts**

**Standard 5.2****b**

Students can compare and order fractions, mixed numbers, and/or decimals in a given set, from least to greatest and greatest to least.**Standard 5.2**

**b**

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **

**Rich Tasks:**

The Tallest Bean Plant by Math Strength

The Greatest Leftovers by Math Strength

**Games/Tech:**

**Games/Tech:**

- Decio↗
- Pecking Order: Fractions and Decimals↗
- Comparing Number Values↗
- Desmos 5.2b Comparing and Ordering Fractions and Decimals 5.2b Fraction/Decimal Clothes Line↗

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## Standard 5.4

**S****tandard**** 5.4 **Create and solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of whole numbers.** **

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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).

Students will develop a relationship between inverse relationships/operations and determine which operation needs to be utilized.

Students can describe their answer and/or the meaning of the remainder and how it affects the solution.

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 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 will use units to represent 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?

**Student Strengths**

**Student Strengths**

Students can solve a variety of computation problems with a chosen strategy, determine if a total is known or unknown, and restate a single-step word problem in their own words.

**Bridging Concepts**

Students can solve a variety of computation problems with an efficient strategy. Students can make connections between various computation strategies and determine how they are similar/different.**Bridging Concepts**

**Standard 5.4**

Students can create and solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of whole numbers.**Standard 5.4**

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **

**Rich Tasks:**

**Games:**

**Games:**

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## Standard 5.5a

**S****tandard**** 5.****5a** Estimate and determine the product and quotient of two numbers involving decimals

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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 will use estimation and rounding in order to determine where to place the decimal in the product or quotient.

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

The student can interpret 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 answer? 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?

**Student Strengths**

**Student Strengths**

Students can add and subtract decimals, multiply with whole numbers, and divide with whole numbers.

**Bridging Concepts**

**Bridging Concepts**

Students can use reasoning and/or estimation to determine placement of the decimal in a multiplication problem. Students can use partial products for multiplication and partial quotients for division as supports.

**Standard ****5.5a**

**Standard**

**5.5a**

Students can estimate and determine the product and quotient of two numbers involving decimals.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

- Number String by Math Strength↗

**Rich Tasks: **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

Desmos 5.5a Discovering Base 10 Patterns: Decimal Division 5.5a Decimal Hopping: Connecting Models and Equations 5.5a Demonstration: Decimal Multiplication Area Model↗

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## Standard 5.5b

**S****tandard**** 5.****5b** Create and solve single-step and multistep practical problems involving addition, subtraction, and multiplication of decimals, and create and solve single-step practical problems involving division of decimals.

**(Pull down for more)**

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 can check their work for reasonableness based on estimating and rounding.

The student can use efficient strategies to add, multiply, subtract, and divide with decimal numbers.

The student can interpret a multistep problem and determine 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?

**Student Strengths**

**Student Strengths**

Students can solve single step practical problems involving addition, subtraction, multiplication and division of whole numbers.

**Bridging Concepts**

**Bridging Concepts**

Students can solve single step practical problems involving addition, subtraction, and multiplication of decimals.

Students can check for reasonableness of answers by estimating.

**Standard 5.****5b**

**Standard 5.**

**5b**

Students can create and solve single-step and multistep practical problems involving addition, subtraction, and multiplication of decimals, and create and solve single-step practical problems involving division of decimals.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 5.6a

**S****tandard**** 5.****6a** Solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed numbers

**(Pull down for more)**

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?

**Student Strengths**

**Student Strengths**

Students can add and subtract fractions with like denominators up to 1 whole. Students can add and subtract mixed numbers with like-denominators. Students can use manipulatives and/or models to find the answer.

**Bridging Concepts**

**Bridging Concepts**

Students can add and subtract fractions with unlike denominators to 1 whole. Students can add and subtract fractions with unlike denominators to 2 wholes. Students can add and subtract with mixed numbers and regrouping by renaming.

**Standard 5.****6a**

**Standard 5.**

**6a**

Students can solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed numbers.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

Fraction Tracks↗ Online Version

Fraction Tracks↗ to 1 and 2 wholes

Desmos 5.6a The Hike↗

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## Standard 5.6b

**S****tandard**** 5.****6b** Solve single-step practical problems involving multiplication of a whole number, limited to 12 or less, and a proper fraction, with models.

**(Pull down for more)**

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?

**Student Strengths**

**Student Strengths**

Students can represent equivalent fractions through twelfths, using region/area models, set models, and measurement/length models.

**Bridging Concepts**

**Bridging Concepts**

Students can use a set model to determine the fraction of a whole using only unit fractions.

**Standard 5.****6b**

**Standard 5.**

**6b**

Students can solve single-step practical problems involving multiplication of a whole number, limited to 12 or less, and a proper fraction, with models.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 5.7

**S****tandard**** 5.7** Simplify whole number numerical expressions using the order of operations.

**(Pull down for more)**

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 is a convention utilized to give 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:**

Students can explain and reason that inverse operations (addition/subtraction; multiplication/division) have equal importance when simplifying an expression.

Students can be given a whole number based numerical expression involving more than one operation, and describe which operation is completed first, second, etc., and why.

Students can explain that the order of operations is a convention used so that all mathematicians can come to the same conclusion when solving an expression.

Students can accurately show their work step-by-step in a logical manner to be able to check their 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?

**Student Strengths**

**Student Strengths**

Students can apply strategies, including place value and the properties of addition to determine the sum or difference of two whole numbers, each 999,999 or less.

Students can apply strategies, including place value and the properties of multiplication and/or addition, to determine the product of two whole numbers when both factors have two digits or fewer.

**Bridging Concepts**

**Bridging Concepts**

Students can identify the relationship between multiplication and division. Students can complete a number string in order from left to right.

**Standard 5.7**

**Standard 5.7**

Students can simplify whole number numerical expressions using the order of operations.

**Full Module with Instructional Tips & Resources:**

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

Today’s Date- Every day there is an expression on the board that equals the day’s date. Then, students must create another expression that equals that date.

**Rich Tasks: **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 5.8a

**S****tandard**** ****5.8a** Solve practical problems that involve perimeter, area, and volume in standard units of measure**.**

**(Pull down for more)**

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. For example this rectangular prism has a bottom row area of 4x3=12 cubes. The second layer also is created by a layer of 12 cubes.

**Important Assessment Look Fors:**

Students can differentiate between perimeter, area, and volume.

Students recognize the relationship between triangles and rectangles.

Students use an efficient strategy to find the perimeter (add all sides or 2L + 2W), area (L x W) and volume.

Students can determine 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?

**Student Strengths**

**Student Strengths**

Students can add multiple strings of numbers; define a rectangle as a quadrilateral with 4 sides with 4 right angles and opposite sides congruent; find the perimeter and area of a rectangle; find the product of 3 whole numbers; and describe volume as cubic units.

**Bridging Concepts**

**Bridging Concepts**

Students can find the total length of 4 numbers; create a rectangle with given dimensions, then find the perimeter and area of that rectangle; relate perimeter and area by finding various rectangles that fit the criteria based on perimeter/area; find the volume of a rectangular prism; and relate the formula for the volume of a rectangular prism to that of length x width x height in a 3-dimensional box using cubes.

**Standard ****5.8A**

**Standard**

**5.8A**

Students can solve practical problems that involve perimeter, area, and volume in standard units of measure.

## Standard 5.8b

**S****tandard**** 5.8b** Differentiate among perimeter, area, and volume and identify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation.

**(Pull down for more)**

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. For example this rectangular prism has a bottom row area of 4x3=12 cubes. The second layer also is created by a layer of 12 cubes.

**Important Assessment Look Fors:**

The student can describe a practical situation where perimeter, area, and volume are appropriate measures to use and justify 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?

**Student Strengths**

**Student Strengths**

Students can describe perimeter as the distance around a polygon, describe area as space covered inside a polygon, and describe volume as the space inside a 3-dimensional figure.

**Bridging Concepts**

**Bridging Concepts**

Students can describe volume of a three-dimensional figure as a measure of capacity and is measured in cubic units.

**Standard ****5.8B**

**Standard**

**5.8B**

Students can differentiate among perimeter, area, and volume and identify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation.

**Full Module with Instructional Tips & Resources:**

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games:**

**Games:**

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## Standard 5.9b

**S****tandard**** 5.****9****b** Solve practical problems involving length, mass, and liquid volume using metric units.

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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).

**Student Strengths**

**Student Strengths**

Students have experience using a ruler and scales to measure a variety of objects. Students can compare objects using customary units.

**Bridging Concepts**

**Bridging Concepts**

Students can measure an object in centimeters and meters and know their relationship. Students can compare objects using metric units.

**Standard 5.****9****B**

**Standard 5.**

**9**

**B**

Students can solve practical problems involving length, mass, and liquid volume using metric units.

**Full Module with Instructional Tips & Resources:**

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 5.16b

**S****tandard**** 5.****16****b** Interpret data represented in line plots and stem-and-leaf plots.

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Understanding the Learning Trajectory

**Big Ideas:**

By the end of Grade 5, students should be comfortable making line plots for measurement data and analyzing data shown in the form of a line plot. (Common Core Progression, 2019, p. 11)

The emphasis in all work with statistics should be on the analysis of the data and the communication of the analysis, rather than on a single correct answer. Data analysis should include opportunities to describe the data, recognize patterns or trends, and make predictions.

There are two types of data: categorical and numerical. Categorical data can be sorted into groups or categories while numerical data are values or observations that can be measured. For example, types of fish caught would be categorical data while weights of fish caught would be numerical data.

**Important Assessment Look Fors:**

The student can interpret data by making observations from line plots and describe the characteristics of the data and describing the data as a whole.

The student can interpret data by making observations from stem-and-leaf plots and describe the characteristics of the data and describing the data as a whole.

The student can interpret data by making inferences from line plots and stem-and-leaf plots.

The student can make generalizations based on the observations of the data.

**Purposeful Questions: **

Can you describe the data as a whole explaining patterns or trends that you see?

What does each X represent in the line plot, and what does it mean when an X appears multiple times above a number on the number line?

What is the difference between a stem and a leaf?

Which graph do you think better represents the data and why?

**Student Strengths**

**Student Strengths**

Students can interpret data by making inferences from bar graphs and line graphs. Students can interpret the data to answer the question posed, and compare the answer to the prediction.

**Bridging Concepts**

**Bridging Concepts**

Students can create a line plot and then use this data to create a stem-and-leaf plot.

**Standard 5.****16****B**

**Standard 5.**

**16**

**B**

Students can interpret data represented in line plots and stem-and-leaf plots.

**Full Module with Instructional Tips & Resources:**

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

Think/Pair/Share- Display a stem and leaf plot with real-world data

**Rich Tasks: **** **

**Rich Tasks:**

**Games:**

**Games:**

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## Standard 5.17a

**S****tandard**** 5.****17a** Given a practical context, describe mean, median, and mode as measures of center

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Understanding the Learning Trajectory

**Big Ideas:**

Students see the mean as a “leveling out” of the data in the sense of a unit rate. In this “leveling out” interpretation, the mean is often called the “average” and can be considered in terms of “fair share.” (Common Core Writing Team, 2019, p.6-8 statistics and probability). It can also be discussed as a balance point.

Mean, median, and mode are all measures of center but depending on the data each can be argued as better representation of the data.

The median is the middle value of a data set in ranked order. Given an odd number of pieces of data, the median is the middle value in ranked order. If there is an even number of pieces of data, the median is the arithmetic average of the two middle values.

Mean represents a fair share concept of the data. Dividing the data constitutes a fair share. This idea of dividing as sharing equally should be demonstrated visually and with manipulatives to develop the foundation for the arithmetic process. (Curriculum Framework, pg 35)

**Important Assessment Look Fors:**

Student describes why the mean, median, and mode are all measures of center.

Student describes how to find the mean, median, and mode.

Student can justify why the practical situation represents the mean, median, or mode.

Student explains how finding the mean is similar to finding a fair share.

**Purposeful Questions: **

What is the difference between each measure of center?

Why are the mean, median, and mode all considered measures of center?

Why did you choose the mean/median/mode to describe this situation?

**Student Strengths**

**Student Strengths**

Students can identify, describe, create, and extend patterns found in objects, pictures, numbers, and tables. Students can describe trends they see in data (Ex. “It is increasing/decreasing.).

**Bridging Concepts**

**Bridging Concepts**

Students can differentiate between the different measures of center and describe the mean as a fair share.

**Standard 5.****17a**

**Standard 5.**

**17a**

Given a practical context, students can describe mean, median, and mode as measures of center.

**Full Module with Instructional Tips & Resources:**

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games:**

**Games:**

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## Standard 5.17d

**S****tandard**** 5.****17d** Determine the mean, median, mode, and range of a set of data

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Understanding the Learning Trajectory

**Big Ideas:**

In Mathematics, the context of data is important. As the Guidelines for Assessment and Instruction in Statistics Education Report notes, “data are not just numbers, they are numbers with a context. In mathematics, context obscures structure. In data analysis, context provides meaning.” (Common Core Writing team, 2019 K–3, Categorical Data; Grades 2–5, Measurement Data*, pg. 3)

Statistics is the science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from data.

Students need to learn more than how to identify the mean, median, mode, and range of a set of data. They need to build an understanding of what the measure tells them about the data, and see those values in the context of other characteristics of the data in order to best describe the results.

**Important Assessment Look Fors:**

The student can determine the mean of a group of numbers representing data from a given context as a measure of center. Students will notice a relationship between outliers and how they change or shift the mean and begin to see that mean is a balance point.

The student can determine the median of a group of numbers representing data from a given context as a measure of center including when there is an even number of data points.

The student can determine the mode of a group of numbers representing data from a given context as a measure of center.

The student can determine the range of a group of numbers and discuss why spread is important to the data’s larger picture.

**Purposeful Questions: **

How did you determine the mean/median? What does the mean/median tell you about the data?

When does your data have or not have a mode? When does it have more than one mode?

What does the range tell us about your data?

Which measure of center (median, mode, or mean) represents this data best? Why?

Do you have any outliers? Why or why not? If so, how does including or removing the outlier from the data affect the mean?

**Student Strengths**

**Student Strengths**

Students can put numbers in order from least to greatest; identify the greatest and least value out of a set of numbers; and identify the whole numbers and decimals that appear the most in a group of numbers.

**Bridging Concepts**

**Bridging Concepts**

Students can find the middle whole number and/or decimal in a set of numbers with an even or odd amount and can find the average or mean of a set of numbers.

**Standard 5.****17d**

**Standard 5.**

**17d**

Students can determine the mean, median, mode, and range of a set of data.

**Full Module with Instructional Tips & Resources:**

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 5.18

**S****tandard**** 5.****18** Identify, describe, create, express, and extend number patterns found in objects, pictures, numbers and tables

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Understanding the Learning Trajectory

**Big Ideas:**

In Mathematics, students learn to see a number as composed of its base-ten units (MP.7). They learn to use this structure and the properties of operations to reduce computing a multi-digit sum, difference, product, or quotient to a collection of single-digit computations in different base-ten units. (ARIZONA, pg. 4)

Mathematical relationships exist in patterns. There are an infinite number of patterns.

Patterns and functions can be represented in many ways and described using words, tables, and symbols.

**Important Assessment Look Fors:**

The student identifies, creates, describes, and extends patterns using concrete materials, number lines, tables, or pictures.

The student can describe and express the relationship found in patterns, using words, tables, and symbols.

The student can solve practical problems that involve identifying, describing, and extending single-operation input and output rules.

The student can identify the rule in a single-operation numerical pattern found in a list or table.

**Purposeful Questions: **

Does the pattern appear to be increasing or decreasing?

What strategies did you use to determine the rule?

Does your rule work for each consecutive number?

How does determining the rule help you understand what is happening and predict what term comes next?

How is this numerical pattern similar to this pattern that is found in the table?

If this pattern continues, what would be the 5th term?

**Student Strengths**

**Student Strengths**

Students can identify and describe patterns, using words, objects, pictures, numbers, and tables. Students can create and extend patterns using objects, pictures, numbers, and tables.

**Bridging Concepts**

**Bridging Concepts**

Students can identify the rule in a single-operation numerical pattern found in a list or table, limited to addition, subtraction, and multiplication of whole numbers.

**Standard 5.18**

**Standard 5.18**

Students can identify, describe, create, express, and extend number patterns found in objects, pictures, numbers and tables (with whole numbers, decimals and fractions).

**Full Module with Instructional Tips & Resources:**

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 5.19a

**S****tandard**** 5.1****9a** Investigate and describe the concept of variable

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Understanding the Learning Trajectory

**Big Ideas:**

According to Blanton’s research on early algebra, in order to overcome the difficulty students have with variable (and variable notation) in formal algebra in later grades , she cites the need for children to begin to engage with mathematical situations, such as the revised task given here, in which they are not operating entirely on knowns: “If Marta has a box of candies and her mother gives her 6 more pieces, how would you describe the number of pieces she has all together?” (This type of task is adapted from Carraher, Schliemann, & Schwartz, 2008). Although the transformation of this task is simple (making the known number of candies in Marta’s box unknown), the opportunity for young children to mathematicize a problem where an unknown exists as part of the formulation of the problem is critical. This helps students recognize the unknown in a situation or a word problem.

Students are introduced to the concept of a variable (presented as boxes, or other symbols) as a representation of an unknown quantity in earlier grades and are substituted with a letter as they advance to upper grades.

**Important Assessment Look Fors:**

The student can identify the variable in a given expression.

The student can use words to describe the unit the variable represents.

**Purposeful Questions: **

What unit/unknown does the variable represent? How do you know?

Describe what a variable is in your own words.

**Student Strengths**

**Student Strengths**

Students can identify ways to solve a problem with an unknown number and students can write an equation.

**Bridging Concepts**

**Bridging Concepts**

Students can write an equation with an unknown and represent the unknown with a symbol.

**Standard 5.1****9a**

**Standard 5.1**

**9a**

Students can investigate and describe the concept of variable.

**Full Module with Instructional Tips & Resources:**

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games:**

**Games:**

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## Standard 5.19b

**S****tandard**** 5.1****9b** Write an equation to represent a given mathematical relationship, using a variable

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Understanding the Learning Trajectory

**Big Ideas:**

Mathematical situations and structures can be translated and represented abstractly using variables, expressions, and equations. (Charles, 2005)

There are many different problem types that can be represented algebraically (join, separate, comparison, equal groups, area, etc.).

The equal sign represents an equality of two expressions that balance each other out.

An unknown quantity can be represented using a variety of variables (letters, numbers, boxes, etc.).

**Important Assessment Look Fors:**

The student identifies the unknown number in a problem and assigns a variable to it.

The student writes a correct equation to represent the problem.

The student explains how a written equation matches a verbal description.

**Purposeful Questions: **

How did you determine what operation to use for this problem?

How did you determine what the variable represents?

Were there any visuals (action words) you used to help you picture the problem?

**Student Strengths**

**Student Strengths**

Students can use bar models or other pictures to represent the unknown quantity in a story problem.