Download Grade 5 - Geometry

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…
Students will understand…
Students will be able to…
1. Number line
1. Ordered pairs are used to locate specific
1. use perpendicular lines to construct a
2. Axes
locations on a coordinate plane.
coordinate system that includes an origin
3. Coordinate system
2. locate and name given points using
4. Intersections
ordered pairs
5. Lines
3. understand that the first number in an
6. Origin
ordered pair (x-coordinate), indicates how
7. Point
far to travel on the x-axis
8. Plane
4. understand that the second number in an
9. Ordered pair
ordered pair (y-coordinate), indicates how
10. X-axis, y-axis
far to travel on the y-axis
11. coordinates
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Point
1. Coordinate values are represented in the
1. reference real-world and mathematical
2. Quadrant
real-world through maps, charts, etc.
problems, including the traveling from one
3. Coordinate plane
point to another and identifying the
4. Coordinate values
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Attributes
1. Logical relationships between categories
1. Understand that every category has
2. Category
and connecting subcategories of 2-D
attributes and that any sub-category in that
3. 2-dimensional figures
figures.
category will have those same attributes
4. Sub-categories
5.G.4.: Classify 2-dimensional figures in a hierarchy based on properties.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
1. 2-dimensional figures
1. 2-dimensional figures can be classified
2. Hierarchy
into categories based on their properties
3. properties
Students will be able to…
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Standard measurement unit
1. Application of multiplication/division to
1. Convert standard measurement units
2. Customary and Metric measurement
execute real-world problems
within a measurement system
systems
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Mean
1. How mean is affected by data distribution
1. Equally redistribute fractions to find mean
2. Line plot
2. Different displays of data can be used to
2. Use operations on fractions to solve
3. Redistribution
solve problems
problems based on information from the
4. Linear and volume measurement
line plot
5. Benchmark fractions
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Volume
1. Volume must be measured with objects
1. Differentiate between linear and volume
2. Attribute
containing volume (same attributes).
attributes including the tools used to
3. Solid figure
measure
4. Unit cube
5. One cubic unit
5.MD.4.: Measure volumes by counting unit cubes, using cubic centimeters, cubic inches, cubic feet, and improvised units
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Abbreviated terms for measurement
1. Volume can be measured using units with
1. Measure volumes by counting unit cubes,
2. Volume
volume within customary and metric, and
using cubic centimeters, cubic inches,
3. Unit
nonstandard units
cubic feet, and improvised units
4. Cubic (including notations
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Volume
1. Volume can be expressed in various ways
1. Model the formula for volume.
2. Operations of multiplication and division
2. How to find the volume of right
2. Generalize formula for volume from
3. Right rectangular prism
rectangular prisms
modeling
4. Unit cubes
3. How volume applies to real-world
3. Apply associative property to the formula
5. Length
concepts
for finding volume
6. Width
4. Transfer concrete model to abstract
4. Add the volume of two or more solid
7. Height
formula for volume
figures
8. Area
9. Base
10.
11.
12.
13.
14.
15.
Volume formulas
Products
Associative property
Additive
Overlapping/non-overlapping
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.
Students will know…
1. multi-digit number
2. Place value
3. Direction (Left vs. Right)
Grade 5 Enduring Understandings
Students will understand…
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Patterns
1. Numbers are related and compare to one
1. Explain patterns
2. Number of Zeros of the product
another in regard to place value.
2. Multiply by powers of 10
3. Powers of 10
2. There are patterns in the number of zeros
3. Divide by powers of 10
4. Placement of the decimal point.
of the product and quotient when
4. Represent powers of 10 with whole
5. Whole number exponents.
multiplying and dividing by powers of 10.
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Decimals to thousandths
1. numbers to thousandths.
1. Read decimals to thousandths
2. Base ten numerals
2. Write decimals to thousandths
3. Number names
3. Compare decimals to thousandths
4. Expanded form
4. Read and write decimals to thousandths
5. <, >, =
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.
Students will know…
1. Place Value
2. Decimals
Grade 5 Enduring Understandings
Students will understand…
1. How to round decimals to any place.
5.NBT.5.: Fluently multiply multi-digit whole numbers using the standard algorithm.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
1. Standard algorithm for multiplying multi1. Multiplication in multi-digit numbers.
digit whole numbers.
Students will be able to…
1. Use place value understanding to round
decimals to any place.
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Whole number quotients
1. The relationship between the quotient,
1. 1. Find whole number quotients of whole
2. Dividends
dividend, and divisor.
numbers with up to 4 digit dividends and
3. Divisors
2. The relationship between multiplication
two-digit divisors
4. Properties of operations
and division.
For example: 1,323/21 = 63
5. Relationship between multiplication and
2. Use strategies based on place value, the
division (inverses)
properties of operations, and/or the
6. Equations
relationship between multiplication and
7. Rectangular arrays
division.
8. Area models
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. decimals to hundredths
1. How concrete models or drawings relate to 1. Add, subtract, multiply, and divide
2. concrete models
adding, subtracting, multiplying, and
decimals to hundredths.
3. cultural contexts
dividing decimals to hundredths.
2. Use concrete drawings and strategies
4. place value
based on place value, properties of
5. properties of operations
operations, and/or the relationship between
6. relationship between addition and
addition and subtraction.
subtraction (inverse)
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Fractions
1. Every fraction has equivalent fractions that 1. Add and Subtract fractions with unlike
2. Unlike denominators
can be used to add or subtract.
denominators by using equivalent fractions
3. Mixed numbers
with like denominators.
4. Equivalent fractions
5. Equivalent sum
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.
Students will know…
1. Fractions
2. Denominators
3. Benchmark fractions
4. estimate
Grade 5 Enduring Understandings
Students will understand…
1. Fractions relate to a whole.
2. Benchmark fractions can be used to
compare.
Students will be able to…
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?
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
3. fraction
1. Fractions are always division problems.
3. Use visual fraction models to represent
4. numerator
fractions as division of the numerator by
5. denominator
the denominator.
6. mixed numbers
4. Solve word problems involving division of
7. equations
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. multiplication of fractions and whole
1. a/b x c/d = ac/bd
1. Apply understanding of multiplication to
numbers by fractions.
2. Interpret the product (a/b) x q as parts of
multiply a fraction or a whole number by a
2. Product
partition of q into b = parts; equivalently,
fraction.
3. Partition
as a result of a sequence of operations a x
2. Extend previous understanding of
4. Operations
q/b, where q is the whole number
multiplication to multiply a fraction or a
5. Visual fraction models
whole number by a fraction, specifically to
6. Numerator
be able to create a story context for the
7. Denominator
equation.
8. Fractions are division problems
9. Properties of multiplication
10. Relationship between multiplication and
division (inverse).
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. scaling in regards to multiplication using
1. the scaling relationship between
5. Understand that multiplying 2 whole
whole numbers and fractions including
multiplying whole numbers, proper
numbers or a whole number by an
improper and proper fractions
fractions, and improper fractions
improper fraction results in a larger whole
2. factors
product
3. products
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
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
12. Multiplication of fractions and mixed
1. Multiplication can be represented in the
1. Solve real world problems involving
numbers
real world
multiplication of fractions and mixed
13. Equations
numbers
14. Fraction models
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?
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Division of unit fractions by whole
1. Relationship between unit fractions and
2. Divide fractions by whole numbers and
numbers
whole numbers
whole numbers by fractions
2. Division of whole numbers by unit
2. Division means to put into equal groups so 3. Apply the inverse relationship of
fractions
fractions divided by whole numbers result
multiplication to divide fractions. For
in smaller quotients while fractions
example- (1/5) / 3 is the same as (1/5) x
3. *The concept unit fraction is a fraction
divided by fractions result in a larger
(1/3).
that has a one in the denominator. For
quotient.
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
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
8. Parenthesis
2. Beginning order of operations concepts in
2. Evaluate expressions that including
9. Brackets
regards to parenthesis
parenthesis, brackets, or braces
10. Braces
3. Use parenthesis, brackets, or braces to
11. Symbols
write expressions
12. Expressions
13. Evaluate
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
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Expression
1. How to read to interpret expressions
1. Write simple expressions that illustrate
2. Calculations
2. Expressions can be written to describe
mathematical thinking (example:
3. Evaluate
mathematical operations
expressing add 8 and 7, then multiply by
4. Parenthesis
2” as 2 x (8 + 7).
5. Sum
2. Interpret numerical expression without
6. Product
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.
Grade 5 Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Ordered pair
1. Two patterns can be expressed as an x,y
1. Generate two numerical patterns using
2. Plane
relationship
two given rules.
3. Coordinate
2. Patterns can be expressed as linear
2. Identify the pattern of x-coordinate
4. Corresponding terms
functions
3. Identify the pattern of y-coordinate
5. Linear function
3. Linear can be graphed on a coordinate
4. Relate the x and y patterns (for example, if
6. Numerical patterns
plane
the line is “steep” y must be increasing
7. Coordinate plane
faster than x)
5. Form ordered pairs from the patterns and
graph them on a coordinate plane
6. Explain the relationship informally