grade 5 - geometry can geometry be used to solve problems about real-world ... represent real-world...

Grade 5 - Geometry Essential Questions:

1. Why are geometry and geometric figures relevant and important?

2. How can geometric ideas be communicated using a variety of representations?

******(i.e maps, grids, charts, spreadsheets)

3. How can geometry be used to solve problems about real-world situations, spatial relationships, and logical reasoning?

We want students to understand that geometry is all around us in 2 or 3-D shapes. Geometric shapes have certain properties and can be

transformed, compared, measured, and constructed.

5.G.1.: Use a pair of perpendicular number lines called axes to define a coordinate system with the intersection of the line (the origin) arranged to

coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers called its coordinates. Understand that

the first indicates how far to travel from the origin in the direction of one axis and the second number indicates how far to travel in the direction of

the second axis with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x coordinate, y-axis and y

coordinate).

Grade 5 Enduring Understandings

Students will know…

1. Number line

2. Axes

3. Coordinate system

4. Intersections

5. Lines

6. Origin

7. Point

8. Plane

9. Ordered pair

10. X-axis, y-axis

11. coordinates

Students will understand…

1. Ordered pairs are used to locate specific

locations on a coordinate plane.

Students will be able to…

1. use perpendicular lines to construct a

coordinate system that includes an origin

2. locate and name given points using

ordered pairs

3. understand that the first number in an

ordered pair (x-coordinate), indicates how

far to travel on the x-axis

4. understand that the second number in an

ordered pair (y-coordinate), indicates how

far to travel on the y-axis

5.G.2.: represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane and interpret coordinate

values of points in the context of the situation.



1. Point

2. Quadrant

3. Coordinate plane

4. Coordinate values


1. Coordinate values are represented in the

real-world through maps, charts, etc.


1. reference real-world and mathematical

problems, including the traveling from one

point to another and identifying the

coordinates of missing points in geometric

figures, such as squares, rectangles, and

parallelograms.

5.G.3.: Understand that attributes belonging to a category of 2-dimensional figures also belong to all subcategories of that category. For example,

all rectangles have 4 right angles and squares are rectangles, so all squares have 4 right angles.



1. Attributes

2. Category

3. 2-dimensional figures

4. Sub-categories


1. Logical relationships between categories

and connecting subcategories of 2-D

figures.


1. Understand that every category has

attributes and that any sub-category in that

category will have those same attributes

5.G.4.: Classify 2-dimensional figures in a hierarchy based on properties.



1. 2-dimensional figures

2. Hierarchy

3. properties


1. 2-dimensional figures can be classified

into categories based on their properties


1. Classify 2-dimensional figures in a

hierarchy based on properties

Grade 5 - Measurement Essential Questions:

1. How does estimation help you find a reasonable measurement?

2. How do you determine the tool and unit to help you accurately measure?

3. When do you need to measure?

Essential Vocabulary – Customary (measurement system), Metric (measurement system), Unit, Conversion, Line plot, data, benchmark fractions,

mean, interpret, analyze, Volume, Attribute, Solid figure, Unit cube, One cubic unit, Abbreviated terms for measurement (example: cm), Cubic,

Unit, Volume, Operations of multiplication and division, Right rectangular prism, Unit cubes, Length, Width, Height, Area, Base, Volume

formulas, Products, Associative property, Additive, Overlapping/non-overlapping, Know that b = base which is l x w (area of rectangle)

We want students to understand when to measure, what tool and unit to use, and how to use estimation to find a reasonable measurement.

5.MD.1.: Convert among different sized standard, measurement units within a given measurement system (e.g convert 5 cm to 0.05 m) and use

these conversions in solving multistep real-world problems within a cultural context including Montana American Indians.



1. Standard measurement unit

2. Customary and Metric measurement

systems


1. Application of multiplication/division to

execute real-world problems


1. Convert standard measurement units

within a measurement system

2. Use conversions to solve real-world

problems

5.MD.2.: Make a line plot to display a data set of measurements in fractions of a unit (1/4, ½, 1/8). Use operations on fractions for this grade to

solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the

amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

This standard provides a context for students to work with fractions by measuring objects to one-eighth of a unit. This includes length, mass, and

liquid volume. Students are making a line plot of this data to have a visual representation for estimating mean and then using operations with

fractions for further analysis based on data in the line plot.



1. Mean

2. Line plot

3. Redistribution

4. Linear and volume measurement

5. Benchmark fractions


1. How mean is affected by data distribution

2. Different displays of data can be used to

solve problems


1. Equally redistribute fractions to find mean

2. Use operations on fractions to solve

problems based on information from the

line plot

5.MD.3.: Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

A cube with side length one unit, called a “unit cube” is said to have “one cubic unit” of volume and can be used to measure volume

A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.



1. Volume

2. Attribute

3. Solid figure

4. Unit cube

5. One cubic unit


1. Volume must be measured with objects

containing volume (same attributes).


1. Differentiate between linear and volume

attributes including the tools used to

measure

5.MD.4.: Measure volumes by counting unit cubes, using cubic centimeters, cubic inches, cubic feet, and improvised units



1. Abbreviated terms for measurement

2. Volume

3. Unit

4. Cubic (including notations


1. Volume can be measured using units with

volume within customary and metric, and

nonstandard units


1. Measure volumes by counting unit cubes,

using cubic centimeters, cubic inches,

cubic feet, and improvised units

5.MD.5.: Relate volume to the operations of multiplication and division and solve real-world and mathematical problems involving volume.

a. Find within cultural contexts, including Montana American Indians, the volume of a right rectangular prism with whole number side lengths by

packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge length equivalently by multiplying the

height by the area of the base. Represent, three-fold whole number products as volumes, e.g. to represent the associative property of multiplication.

b. Apply the formulas V=l x w x h and V=b x h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths

in the context of solving real-world and mathematical problems.

c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes

of the non-overlapping parts, applying this technique to solve real-world problems.



1. Volume

2. Operations of multiplication and division

3. Right rectangular prism

4. Unit cubes

5. Length

6. Width

7. Height

8. Area

9. Base


1. Volume can be expressed in various ways

2. How to find the volume of right

rectangular prisms

3. How volume applies to real-world

concepts

4. Transfer concrete model to abstract

formula for volume


1. Model the formula for volume.

2. Generalize formula for volume from

modeling

3. Apply associative property to the formula

for finding volume

4. Add the volume of two or more solid

figures

10. Volume formulas

11. Products

12. Associative property

13. Additive

14. Overlapping/non-overlapping

15. Know that b = base which is l x w (area of

rectangle)

Grade 5 – Numbers Base 10 Essential Questions:

1. Why do we use numbers, what are their properties, and how does our number system function?

2. Why do we use estimation and when is it appropriate?

3. What makes a strategy effective and efficient and the solution reasonable?

4. How do numbers relate and compare to one another?

Essential Vocabulary:

We want students to understand that all numbers have value, uses, types, and we use operations and reasonableness to work with them.

5.NBT.1. Recognize that in a multi-digit number a digit in one place represents ten times as much as it represents in the place to its right

and 1/10 of what it represents in the place to its left.



1. multi-digit number

2. Place value

3. Direction (Left vs. Right)


1. Place value. Students will be able to…

1. Recognize meaning of numbers based on

their place value.

5.NBT.2.: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in

the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote

powers of 10.



1. Patterns

2. Number of Zeros of the product

3. Powers of 10

4. Placement of the decimal point.

5. Whole number exponents.


1. Numbers are related and compare to one

another in regard to place value.

2. There are patterns in the number of zeros

of the product and quotient when

multiplying and dividing by powers of 10.


1. Explain patterns

2. Multiply by powers of 10

3. Divide by powers of 10

4. Represent powers of 10 with whole

number exponents

5.NBT.3.: Read, Write, and Compare decimals to the thousandths. Read and write decimals to thousandths using base-ten numerals,

number names, and expanded form.

Compare two decimals to thousandths based on meanings of digits in each place using greater than, less than, and equal to symbols to

record the results of comparisons.



1. Decimals to thousandths

2. Base ten numerals

3. Number names

4. Expanded form

5. <, >, =


1. numbers to thousandths. Students will be able to…

1. Read decimals to thousandths

2. Write decimals to thousandths

3. Compare decimals to thousandths

4. Read and write decimals to thousandths

using based-tens numerals, number names,

and expanded form

5. Compare two decimals to thousandths

using <, >, or = symbols to record results

5.NBT.4.: Use place value understanding to round decimals to any place.



1. Place Value

2. Decimals


1. How to round decimals to any place. Students will be able to…

1. Use place value understanding to round

decimals to any place.

5.NBT.5.: Fluently multiply multi-digit whole numbers using the standard algorithm.



1. Standard algorithm for multiplying multi-

digit whole numbers.


1. Multiplication in multi-digit numbers. Students will be able to…

1. Fluently multiply multi-digit whole

numbers.

5.NBT.6.: Find whole number quotients of whole numbers with up to 4-digit dividends and two-digit divisors, using strategies based on

place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the

calculation by using equations, rectangular arrays, and/or area models.



1. Whole number quotients

2. Dividends

3. Divisors

4. Properties of operations

5. Relationship between multiplication and

division (inverses)

6. Equations

7. Rectangular arrays

8. Area models


1. The relationship between the quotient,

dividend, and divisor.

2. The relationship between multiplication

and division.


1. 1. Find whole number quotients of whole

numbers with up to 4 digit dividends and

two-digit divisors

For example: 1,323/21 = 63

2. Use strategies based on place value, the

properties of operations, and/or the

relationship between multiplication and

division.

For example: 21 x 63 = 1,323 can be (20 x

63) + (1 x 63) = 1,323

3. Illustrate and explain the calculation by

using equations, rectangular arrays, and

area models. (The distributive property)

5.NBT.7.: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings within cultural contexts,

including those of the Montana American Indians, and strategies based on place value, properties of operations, and/or the relationship

between addition and subtraction; relate the strategy to a written method and explain the reasoning used.



1. decimals to hundredths

2. concrete models

3. cultural contexts

4. place value

5. properties of operations

6. relationship between addition and

subtraction (inverse)


1. How concrete models or drawings relate to

adding, subtracting, multiplying, and

dividing decimals to hundredths.


1. Add, subtract, multiply, and divide

decimals to hundredths.

2. Use concrete drawings and strategies

based on place value, properties of

operations, and/or the relationship between

addition and subtraction.

3. Relate the strategy to a written method

4. Explain the reasoning used.

Grade 5 - Fractions Essential Questions:

1. Why do we use numbers, what are their properties, and how does our number system function?

2. Why do we use estimation and when is it appropriate?

3. What makes a strategy effective and efficient and the solution reasonable?

4. How do numbers relate and compare to one another?

Essential Vocabulary –

We want students to understand that all numbers have value, uses, types, and we use operations and reasonableness to work with them.

5.NF.1.: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fraction with equivalent

fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example: 2/3 + 5/4 = 8/12 +

15/12 = 23/12 or in general a/b + c/d = (ad + bc) / bd.



1. Fractions

2. Unlike denominators

3. Mixed numbers

4. Equivalent fractions

5. Equivalent sum


1. Every fraction has equivalent fractions that

can be used to add or subtract.


1. Add and Subtract fractions with unlike

denominators by using equivalent fractions

with like denominators.

5.NF.2.: Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike

denominators, eg. By using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of

fractions to estimate mentally and assess the reasonableness of answers. For example: recognize an incorrect result 2/5 + ½ = 3/7, by

observing that 3/7 < ½.



1. Fractions

2. Denominators

3. Benchmark fractions

4. estimate


1. Fractions relate to a whole.

2. Benchmark fractions can be used to

compare.


1. Solve word problems with addition and

subtraction of fractions.

2. Use visual fraction models and equations.

3. Estimate using benchmark fractions and

number sense of fractions.

5.NF.3.: interpret a fraction as division of the numerator by the denominator. Solve word problems involving division of whole numbers

leading to answers in the form of fractions or mixed numbers e.g. By using visual fraction models or equations to represent the problem.

For example: interpret ¾ as the result of dividing 3 by 4, noting that ¾ multiplied by 4 = 3 and when 3 wholes are shared equally among 4

people each person has a share of size ¾. If 9 people want to a share a 50 pound sack of rice by weight, how many pounds of rice should

each person get? Between what two whole numbers does your answer lie?



3. fraction

4. numerator

5. denominator

6. mixed numbers

7. equations


1. Fractions are always division problems.


3. Use visual fraction models to represent

fractions as division of the numerator by

the denominator.

4. Solve word problems involving division of

whole numbers where the answer is a

fraction or mixed number.

5.NF.4.: Apply and extend previous understandings of multiplication to multiply a fraction or a whole number by a fraction.

a- Interpret the product (a/b) x q as parts of partition of q into b = parts; equivalently, as a result of a sequence of operations a x q/b.

For example, use a visual fraction model to show (2 /3) x 4 = 8/3, and create a story context for this equation within cultural

contexts, including those of Montana Native Americans; and do the same with (2/3) x( 4/5) = 8/15. (In general,).

b- Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths

and show that the area is the same as could be found by multiplying the side lengths. Multiply fractional side lengths to find areas

of rectangles and represent fraction products as rectangular areas.



1. multiplication of fractions and whole

numbers by fractions.

2. Product

3. Partition

4. Operations

5. Visual fraction models

6. Numerator

7. Denominator

8. Fractions are division problems

9. Properties of multiplication

10. Relationship between multiplication and

division (inverse).


1. a/b x c/d = ac/bd

2. Interpret the product (a/b) x q as parts of

partition of q into b = parts; equivalently,

as a result of a sequence of operations a x

q/b, where q is the whole number


1. Apply understanding of multiplication to

multiply a fraction or a whole number by a

fraction.

2. Extend previous understanding of

multiplication to multiply a fraction or a

whole number by a fraction, specifically to

be able to create a story context for the

equation.

5.NF.5.: Interpret multiplication as scaling (resizing), by;

a- Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the

indicated multiplication

b- Explaining why multiplying a given number by a fraction greater than one results in a product greater than the given number

(recognizing multiplication by whole numbers greater than one as a familiar case); explaining why multiplying a given number by

a fraction less than one results in a product smaller than the given number; and relating the principle of fraction equivalence a/b=

(n x a) / (n x b) to the effect of multiplying a/b by 1.



1. scaling in regards to multiplication using

whole numbers and fractions including

improper and proper fractions

2. factors

3. products


1. the scaling relationship between

multiplying whole numbers, proper

fractions, and improper fractions


5. Understand that multiplying 2 whole

numbers or a whole number by an

improper fraction results in a larger whole

product

6. Understand that multiplying proper

fractions by whole numbers or improper

fractions result in a smaller product

5.NF.6.: Solve real-world problems involving multiplication of fractions and mixed numbers, e.g. by using visual fraction models and

equations to represent the problem within cultural contexts, including those of Montana American Indians



12. Multiplication of fractions and mixed

numbers

13. Equations

14. Fraction models


1. Multiplication can be represented in the

real world


1. Solve real world problems involving

multiplication of fractions and mixed

numbers

2. Use visual fraction models or equations

3. Represent the problem within cultural

contexts

5.NF.7.: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit

fractions.

a- Interpret division of a unit fraction by a nonzero whole number, and compute such quotients. For example, create a story within

cultural contexts including those of MAI, for (1/3) / 4, and use a visual fraction model to show the quotient. Use the relationship

between multiplication and division to explain that (1/3)/4 = 1/12 because (1/12) x 4 = 1/3.

b- Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story within cultural

contexts, including those of MAI, for 4 / (1/5), and use a visual fraction model to show the quotient. Use the relationship between

multiplication and division to explain that 4 / (1/5) = 20 because 20 x (1/5) =4.

c- Solve real-world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit

fractions e.g. by using visual fraction models and equations to represent the problem. For example, how much chocolate will each

person get if 3 people share half a pound of chocolate equally? How many 1/3 cup servings are in 2 cups of raisins?



1. Division of unit fractions by whole

numbers

2. Division of whole numbers by unit

fractions

3. *The concept unit fraction is a fraction

that has a one in the denominator. For

example, the fraction 3/5 is 3 copies of the

unit fraction 1/5. 1/5 + 1/5 + 1/5 = 3/5 =

1/5 x 3 or 3 x 1/5


1. Relationship between unit fractions and

whole numbers

2. Division means to put into equal groups so

fractions divided by whole numbers result

in smaller quotients while fractions

divided by fractions result in a larger

quotient.


2. Divide fractions by whole numbers and

whole numbers by fractions

3. Apply the inverse relationship of

multiplication to divide fractions. For

example- (1/5) / 3 is the same as (1/5) x

(1/3).

Grade 5 – Algebraic Thinking Essential Questions:

1. How do you use patterns to understand mathematics and model situations?

2. What is algebra?

3. How are the horizontal and vertical axes related?

4. How do algebraic representations relate and compare to one another?

5. How can we communicate and generalize algebraic relationships?

Essential Vocabulary – Parenthesis, Brackets, Braces, Symbols, Expressions, Evaluate, Calculations, Sum, Product, Ordered pair, Plane,

Coordinate, Corresponding terms, Linear function, Numerical patterns, Coordinate plane

We want students to understand how we use patterns and relationships of algebraic representations to generalize, communicate, and

model situations in mathematics.

5.OA.1.: Use parenthesis, brackets, or braces in numerical expressions and evaluate expressions with these symbols.



8. Parenthesis

9. Brackets

10. Braces

11. Symbols

12. Expressions

13. Evaluate


2. Beginning order of operations concepts in

regards to parenthesis


2. Evaluate expressions that including

parenthesis, brackets, or braces

3. Use parenthesis, brackets, or braces to

write expressions

5.OA.2.: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For

example, express the calculation “add 8 and 7, then multiply by 2” as 2 x (8 + 7). Recognize that 3 times (18,932 + 921) is 3 times as large as 18,

932 + 921, without having to calculate the indicated sum or product.

5.OA.2. Benchmark



1. Expression

2. Calculations

3. Evaluate

4. Parenthesis

5. Sum

6. Product


1. How to read to interpret expressions

2. Expressions can be written to describe

mathematical operations


1. Write simple expressions that illustrate

mathematical thinking (example:

expressing add 8 and 7, then multiply by

2” as 2 x (8 + 7).

2. Interpret numerical expression without

evaluating

5.OA.3.: Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered

pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “add

three” and the starting number zero, and given the rule “add six” and the starting number zero generate terms in the resulting sequences, and

observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Extends the work from Fourth Grade, where students generate numerical patterns when they are given one rule. In Fifth Grade, students are given

two rules and generate two numerical patterns. The graphs that are created should be line graphs to represent the pattern. This is a linear function

which is why we get the straight lines. The Days are the independent variable, Fish are the dependent variables, and the constant rate is what the

rule identifies in the table.



1. Ordered pair

2. Plane

3. Coordinate

4. Corresponding terms

5. Linear function

6. Numerical patterns

7. Coordinate plane


1. Two patterns can be expressed as an x,y

relationship

2. Patterns can be expressed as linear

functions

3. Linear can be graphed on a coordinate

plane


1. Generate two numerical patterns using

two given rules.

2. Identify the pattern of x-coordinate

3. Identify the pattern of y-coordinate

4. Relate the x and y patterns (for example, if

the line is “steep” y must be increasing

faster than x)

5. Form ordered pairs from the patterns and

graph them on a coordinate plane

6. Explain the relationship informally

grade 5 - geometry can geometry be used to solve problems about real-world ... represent real-world...

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