Number and Operations Grade 5 Mathematics Enduring … · 2018-03-30 · Number and Operations...

Number and Operations Grade 5 Mathematics Enduring Understanding 1: Relationships among numbers and number systems form the foundations of number sense and mathematics communication.

Systems Discipline Level Concept: Relationships

ACPS 2014-‐15 DRAFT

Essential Standards in grade band 3-5 Develop strategies for judging the relative size of numbers, identify classes of numbers and examine their properties

Performance Indicators Assessment Samples – SOL/Blooms Vocabulary All students will: � understand our number

system is a pattern of tens where each place is ten times the value of the place to its right.

� understand that decimals are written as an extension of the place value system.

� understand decimals and fractions represent the same relationships, parts to a whole, but are written in two different forms.

Knowledge/Comprehension Level: � Cut out each fraction square and sort them into the appropriate group.

Application/Analysis Level: � Label the following points on the given number line:

Synthesis/Evaluation Level:

decimal point

tenths

hundredths

thousandths

numerator

denominator

common denominator

simplest form

Number and Operations Grade 5 Mathematics Enduring Understanding 1: Relationships among numbers and number systems form the foundations of number sense and mathematics communication.


ACPS 2014-‐15 DRAFT

� Determine which fraction has the greatest value. Justify your answer in two different ways.

3 , 3 4 5

SOL:

5.1 The student, given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth.

5.2 The student will a) recognize and name fractions in their equivalent decimal form and vice versa; and b) compare and order fractions and decimals in a given set from least to greatest and greatest to least.

5.3 The student will a) identify and describe the characteristics of prime and composite numbers; and

b) identify and describe the characteristics of even and odd numbers.

Number and Operations Grade 5 Mathematics Enduring Understanding 8: Patterns and relationships among operations are essential to making estimates and computing fluently.

Discipline Level Concept: Patterns Change and Interactions

Essential Standard grade band 3-5: Understand the meaning of, and relationships between addition and subtraction, and multiplication and division.

Performance Indicators Assessment Samples – SOL/Blooms Vocabulary All students will: � understand that visual models,

benchmarks, and equivalent forms are essential to solving computations involving fractions and decimals.

� know that multiplication and division are inversely related.

� know that fractions are another way of representing division.

� be able to add or subtract 2 fractions with different denominators.

Knowledge/Comprehension Level: � A number has two digits. Both digits are even. The digit in the tens place is

greater than the digit in the ones place. The ones digit is not in the three times table. The tens digit is not double the ones digit. The sum of the two digits is a multiple of five. Which number am I?

18 86 42 46 80 22 83

Application/Analysis Level: � How many different ways can you make your calculator show 12.5 without

pressing the decimal point button? Show at least 3 ways.

Synthesis/Evaluation Level: � How could you calculate 23 x 21 if the 2 button on your calculator is broken?

front-end estimation

reasonableness

factors

sum

difference

product

quotient

addend

divisor

dividend

numerator

denominator

decimal point

digit

remainder

divisible

least common multiple

greatest common factor

tenths

hundredths

thousandths

SOL:

Number and Operations Grade 5 Mathematics Enduring Understanding 8: Patterns and relationships among operations are essential to making estimates and computing fluently.

Discipline Level Concept: Patterns Change and Interactions

5.1 The student, given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth.

5.2 The student will a) recognize and name fractions in their equivalent decimal form and vice versa; and b) compare and order fractions and decimals in a given set from least to greatest and greatest to least.

5.3 The student will a) identify and describe the characteristics of prime and composite numbers; and b) identify and describe the characteristics of even and odd numbers.

Supporting SOL:

Geometry and Measurement Strand Grade 5 Mathematics Enduring Understanding 2: Spatial relationships can be described using coordinate geometry and other representational systems.


Number of Sides

Name of Polygon

Sketch of Polygon

1

2

3

4

5

6

7

8

Essential Standards in grade band 3-5 Use a coordinate system to aid in describing location and movement in all four quadrants of the Cartesian plane

Performance Indicators

Assessment Samples – SOL/Blooms Vocabulary

All students will:

� understand that coordinate systems specify locations and describe paths.

Knowledge/Comprehension Level: � Using the descriptions on the chart, name the shape and draw it on the grid paper.

“Picturing Polygons”, Investigations in Number, Data, and Space

coordinate system

quadrant

plane

rotate

translate

reflection

horizontal

vertical parallel

lines vertex,

vertices



Application/Analysis Level: � Do you think any of these shapes are impossible to draw? If so, pick one and write about how

you can prove it is impossible. “Picturing Polygons”, Investigations in Number, Data, and Space



Synthesis/Evaluation Level: � Create side and angle descriptions of one quadrilateral that could be drawn and one quadrilateral

that could not be drawn. “Picturing Polygons”, Investigations in Number, Data, and Space

SOL: 5.15e Supporting SOL:

Supporting Skills and Processes:

Geometry and Measurement Strand Grade 5 Mathematics Enduring Understanding 3: Attributes of objects can be measured using processes and quantified units, and using appropriate techniques, tools, and formulas.

Systems Discipline Level Concept: Quantifying Representation

Essential Standards in grade band 3-5: Develop and deepen understanding of what it means to measure an object: identifying an attribute to be measured, choosing an appropriate unit, and comparing that unit to the object being measured

Performance Indicators Assessment Samples – SOL/Blooms Vocabulary All students will: � understand benchmarks

can be used to estimate and measure weight/mass, liquid volume, length, size of angles, and time.

� be able to measure in both metric and customary units to the nearest appropriate unit using a variety of measuring instruments.

� be able to indentify, describe, and distinguish between area, perimeter, and volume.

� Understand estimations can be used to check the reasonableness of measurements.

Knowledge/Comprehension Level: � Grades 3-5 Mathematics Assessment Sampler p. 121 #5

The car in the picture is 3.5 cm long. About how long is the building?

A 18m B 14m C 10m D 4m

perimeter area

volume

length width

side

height

circle

chord diameter radius

circumference customary unit

inch (in.)

feet (ft.)

yard (yd.) mile (mi.)

ounce (oz.)

pound (lb.) cup (c.)

pint (pt.) quart (qt)

gallon (gal.)

metric unit



Application/Analysis Level:

� On the grid below, draw a rectangle with an area of 12 square units. Justify your answer.

= 1 square unit

Synthesis/Evaluation Level � Create your own measurement system to create a scale drawing of

your classroom. In addition, create tools to use for measuring in your new system.

millimeter (mm)

centimeter (cm) kilometer (km)

meter (m) km gram (g)

kilogram (kg)

milileter (mL) liter (L)

weight mass temperature

Fahrenheit

Celsius square units

right angle

obtuse angle

acute angle

protractor angle ruler

SOL: 5.8a The student will: a) find perimeter, area, and volume in standard units of measure; b) 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; c) identify equivalent measurements within the metric system; d) estimate and then measure to solve problems, using U.S. Customary and metric units; and e) choose an appropriate unit of measure for a given situation involving measurement using U.S. Customary and metric

units.



5.9 The student will identify and describe the diameter, radius, chord, and circumference of a circle. 5.10 The student will determine an amount of elapsed time in hours and minutes within a 24-hour period.

5.11 The student will measure right, acute, obtuse, and straight angles

5.13 The student will measure and draw right, acute, and obtuse angles and triangles, using appropriate tools

Geometry and Measurement Strand Grade 5 Mathematics Enduring Understanding 11:: Analyze characteristics and properties of 2- and 3-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Communication Discipline Level Concept: Reasoning and Justification

Essential Standards in grade band 3-5 Develop clarity and precision in describing properties of geometric objects and classifying into categories by properties

Performance Indicators Assessment Samples – SOL/Blooms Vocabulary All students will:

� understand that the defining properties and symmetry of various plane figures are unique.

� understand that simple plane

figures can be combined to make more complicated figures; and, complicated figures can be subdivided into simple plane figures.

� know that solid figures are

unique in their defining properties.

� be able to describe three

dimensional shapes (solids) using number of faces, edges, corners (vertices), and the shape of the faces.

Knowledge/Comprehension level 1. Choose four 3-Dimensional figures from the set of geometric

solids, and list the characteristics for each figure.

2. Using your ruler, draw a polygon that satisfies these conditions. A. The polygon only has two right angles. B. The polygon only has two congruent sides. C. Label the right angles and the two congruent sides.

Application/Analysis level: � Alan says that if a figure has four sides, it must be a rectangle. Gina

does not agree. Draw two figures that show Gina is correct. Explain how your figure supports Gina’s belief that each four-sided figure is NOT a rectangle.

Synthesis/Evaluation level � Mystery Rings worksheet (NCTM) (Attachment 11,5)

parallel

perpendicular

side

angle

vertex

diagonal

similar

congruent

two-dimensional

polygon

triangle

right triangle acute

triangle obtuse

triangle scalene

triangle isosceles

triangle equilateral

triangle

quadrilateral

rectangle

square

parallelogram

Geometry and Measurement Strand Grade 5 Mathematics Enduring Understanding 11:: Analyze characteristics and properties of 2- and 3-dimensional geometric shapes and develop mathematical arguments about geometric relationships


kite

trapezoid

rhombus

three-dimensional

face

edge

pyramid

prism

cylinder

cone

cube

SOL: 5.13 The student, using plane figures (square, rectangle, triangle, parallelogram, rhombus, and trapezoid), will a) develop definitions of these plane figures; and b) investigate and describe the results of combining and subdividing plane figures.

Supporting SOL 5.8 The student will describe and determine the perimeter of a polygon and the area of a square, rectangle, and right triangle, given the appropriate

measures. 5.9 The student will identify and describe the diameter, radius, chord, and circumference of a circle.

5.13 The student will measure and draw right, acute, and obtuse angles and triangles, using appropriate tools. 5.14 The student will classify angles and triangles as right, acute, or obtuse.

Supporting Skills and Processes: Construct models of 2- and 3-dimensional figures using a variety of resources to include technology and manipulatives


Geometry and Measurement Strand Grade 5 Mathematics Enduring Understanding 12: Transformations, symmetry, and spatial reasoning can be used to analyze and model mathematical situations.

Essential Standards in grade band 3-5 Identify symmetry and congruency, and describe the results of various transformations of plane figures

Performance Indicators Assessment Samples – SOL/Blooms Vocabulary

All students will:

� be able to predict and describe the results of sliding, flipping, and turning two–dimensional shapes.

� be able to describe a

motion or series of motions that will show that two shapes are congruent.

� be able to describe

rotational and line symmetry in 2- and 3- dimensional shapes and designs.

Knowledge/Comprehension Level: � Make a list of real-life examples of rotation, translations, and reflections.

Application/Analysis Level: Grades 3-5 Mathematics Assessment Sampler p. 110 #25

transformation

translate

rotate

reflection

line of symmetry

rotational symmetry

congruent

similar figure

plane figure


Geometry and Measurement Strand Grade 5 Mathematics Enduring Understanding 12: Transformations, symmetry, and spatial reasoning can be used to analyze and model mathematical situations.

Synthesis/Evaluation Level: � Grades 3-5 Mathematics Assessment Sampler p. 112 #26 followed by

students creating their own rug design with reflect ional symmetry.

SOL: 5.15 The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will b) identify and explore congruent, non-congruent, and similar figures; d) identify and describe a line of symmetry; and e) recognize the images of figures resulting from geometric transformations such as translation (slide), reflection (flip), or rotation (turn).

Data Analysis and Probability Grade 5 Mathematics Enduring Understanding 5: Mathematical models are used to predict and make inferences about data.

Properties and Models Discipline Level Concept: Models

Essential Standards in grade band 3-5: Propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions or predictions

Performance Indicators

Assessment Samples – SOL/Blooms Vocabulary

All students will: � understand

conclusions and predictions can be made based on displays of data.

Knowledge/Comprehension level outcomes

sample space

single event

predict/prediction

impossible

unlikely

equally likely

likely/likelihood

certain

tree diagram

bar graph

stem-and-leaf plot

line graph

categorical data

ascending order

descending order

vertical axis



Application /Analysis level Directions: Look at the line graph below and decide what this might be a graph about. Explain your reasoning and give it a title.

horizontal axis

70 60 50 40 30 20 10

0

1. What is this a graph of? Explain why.

2. What else could this be a graph of? Explain why.



In the space below, make a chart or graph that shows voting results that fit all four clues. Explain why your chart or graph fits all the clues.

Synthesis/Analysis level

The students in the fourth grade voted for their favorite lunch of the past 3 days.

The three lunches were:

Pizza Cheeseburger Nachos

Here are some clues about the results of the vote. A. 42 students voted B. There was a two-way tie for first place C. Each lunch received at least 5 votes D. No lunch received more than 19 votes

SOL: 5.14 The student will make predictions and determine the probability of an outcome by constructing a sample space.

5.15 The student, given a problem situation, will collect, organize, and interpret data in a variety of forms, using stem-and-leaf plots and line

graphs.

Supporting SOL 5.16c The student will find the mean, median, mode, and range of a set of data.

Supporting Skills and Processes:

Data Analysis and Probability Mathematics Enduring Understanding 6: Data can be collected, organized, and displayed in purposeful ways

Grade 5

Properties and Models Discipline Level Concept: Analysis and Evaluation

Essential Standards in grade band 3-5 Collect, represent, and investigate data and how data collection methods affect the nature of the data set

Performance Indicators Assessment Samples – SOL/Blooms Vocabulary All students will: � be able to display data

using multiple representations.

Knowledge/Comprehension level Virginia SOL 5.18

The 4th, 5th, and 6th grade classes at Elmwood School wanted to earn enough money for new band instruments. They needed to sell a total of 1,000 candy bars. About how many more candy bars do they need to sell?

A 250 B 300 C 350 D 400

data / data set

survey

scientific experiment

poll

questionnaire

bar graph

line graph

pie charts

stem and leaf plots

axis

vertical axis

horizontal axis

scale increment

numerical data

categorical data

change over time


Grade 5


Application /Analysis level

The cafeteria wants to know which lunch is the favorite of the 5th

graders.

spaghetti tacos pizza

Tell how you would collect this information and what kind of graph

you would use to show what you have found.

Synthesis/Evaluation level The students in Mr. Kirby’s class voted for their favorite book of the past 3 months. Babe, The Gallant Pig Sarah, Plain and Tall Stone Fox

Here are some clues about the results of the vote.

a. 34 students voted b. The winner got the most votes, but got fewer than half the votes. c. There was a two-way tie for second place.

Make a chart or graph that shows voting results that fit all three clues. Write an explanation of why you chose the graph that you did.


Grade 5


SOL: 5.15 The student, given a problem situation, will collect, organize, and interpret data in a variety of forms, using stem-and-leaf plots and line graphs.

Supporting Skills and Processes: Formulate questions, conduct surveys, make observations, create bar graphs, create picture graphs, create line plots, create stem and leaf plots, label graphs using titles and keys

Data Analysis and Probability Grade 5 Mathematics Enduring Understanding 7: Various statistical methods can be used to observe, analyze, predict, and make inferences about data.

Essential Understandings Assessment Samples – SOL/Blooms Vocabulary

All students should

! Understand that bar graphs compare categorical data, stem- and-leaf plots list data in a meaningful array, and line graphs show changes over time

! Understand how to propose and

justify conclusions and predictions that are based on displays of data.

! Data analysis should include

opportunities to describe the data, recognize patterns or trends, or make predictions.

Knowledge/Comprehension level

Application /Analysis level


data / data set

survey

scientific experiment

poll

questionnaire

bar graph

line graph

axis

vertical axis

horizontal axis

scale increment

numerical data

categorical data

chart

change over time

SOL: 5.18 The student will, given a problem situation, collect, organize, and display a set of numerical data in a variety of forms using bar graphs, stem-and-leaf plots, and line graphs, to draw conclusions and make predictions.

Supporting Skills and Processes: Formulate questions, conduct surveys, make observations, create bar graphs, create picture graphs, create line plots, create stem and leaf plots, label graphs using titles and keys

In grade band 3-5: Understand the meaning of, and relationships between addition and subtraction, and multiplication and division.


Data Analysis and Probability Mathematics Enduring Understanding 13: Probability and data analysis can be used to make predictions.

Grade 5

Communication Discipline Level Concept: Theory

Essential Standard In grade band 3-5: Describe and discuss the degree of likelihood of events and predict the probability of outcomes of simple experiments

Performance Indicators Assessment Samples – SOL/Blooms Vocabulary All students will: ! understand and apply basic

concepts of probability and make predictions of outcomes of simple experiments.

! understand a sample space

represents all possible outcomes of an experiment.

! understand the measure of

likelihood of an event can be represented by a number from 0 to 1.

Knowledge/Comprehension level A bag contains 3 blue pencils, 4 green pencils, 1 red pencil and 5 yellow pencils. If one is drawn from the bag without looking, what is the probability it is yellow?

Application /Analysis level In the morning Paul is wondering what to wear. Choose one item from each list, identify how many different combinations are possible, and list all of the combinations or make a tree-diagram. Shoes Pants Shirts Sneakers jeans T-shirt Boots shorts sweater

Khakis Hawaiian shirt

collect

organize

display

single event

tree diagram

sample space

results

simple experiment

prediction

stem-and-leaf plot

line graph

bar graph

Data Analysis and Probability Mathematics Enduring Understanding 13: Probability and data analysis can be used to make predictions.

Grade 5

Communication Discipline Level Concept: Theory


There is only one red marble in each of the bags shown below. Without looking, you are to pick a marble out of one of the bags. Which bag would give you the greatest chance of picking the red marble? Explain why.

SOL 5.17 The student will a. Solve problems involving the probability of a single event by using tree diagrams or by constructing a sample space

representing all possible results; b. Predict the probability of outcomes of simple experiments, representing it with fractions or decimals from 0 to 1, and test the

prediction; and c. Create a problem statement involving probability based on information from a given problem situation. Students will not be

required to solve the created problem statement.

Supporting SOL: SOL 5.18 The student will, given a problem situation, collect, organize, and display a set of numerical data in a variety of forms, using bar graphs, stem-and-leaf plots, and line graphs, to draw conclusions and make predictions.

Patterns and Algebra Grade 5 Mathematics Enduring Understanding 4: Situations and structures can be represented, modeled and analyzed using algebraic symbols.


Number of Chirps per Minute

Temperature

144 76º 152 78º 160 80º 168 82º 176 84º

Essential Standards in grade band 3-5 Investigate, describe, and represent various rates of change

Performance Indicators Assessment Samples – SOL/Blooms Vocabulary All students will:

� understand that patterns have structures and can grow and/or change.

� know that mathematical relationships exist in patterns.

� know that a variable is a symbol that can stand for any member of a set of numbers.

Knowledge/Comprehension Level: Which is the variable in the following equation?

N +7 = 12

A. 12 B. 7 C. + D. N


The table below shows how the chirping cricket is related to the temperature outside. For example, a cricket chirps 144 times each minute when the temperature is 76º.

What would be the number of chirps per minute when the temperature outside is 90º if the pattern stays the same? Explain how you figured out your answer.

Synthesis/Evaluation Level:

Equation

Equal

Equivalent

Equality

Open sentence

Variable

Symbol

Number sentence

Pattern

Patterns and Algebra Grade 5 Mathematics Enduring Understanding 4: Situations and structures can be represented, modeled and analyzed using algebraic symbols.


38 36 34 32 30 28 26 24 22 20 18 16 14 12 10

8 6 4 2 0

In what situation could this graph happen? Write a story that would also explain and extend the pattern using mathematical vocabulary.

SOL:5.17 The student will describe the relationship found in a number pattern and express the relationship. 5.18 The student will

a) investigate and describe the concept of variable; b) write an open sentence to represent a given mathematical relationship, using a variable;

Patterns and Algebra Strand Grade 5 Mathematics Enduring Understanding 9: Patterns, relations, and functions can be recognized and understood mathematically.

Change and Interactions Discipline Level Concept: Patterns

Step Number

New tiles Step size

Total So Far

1 3 3 2 3 6 3 3 9 4 ? ? 5 ? ?

Step Number New Tiles (step size) Total So Far 1 1 1 2 3 4 3 5 9 4 ? ?

… ? ? 8 ? ?

Essential Standards in grade band 3-5Investigate numerical and geometric patterns and express them mathematically in words and symbols

Performance Indicators Assessment Samples – SOL/Blooms Vocabulary All students will: � understand that patterns

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

� understand how a pattern grows and changes.

� understand that mathematical relationships exist in patterns.

Knowledge/Comprehension Level: � I am building a pattern with tiles. I have represented the first three steps in the chart. What

will the next two steps be? Put your answers in the chart.


This chart is an example of a growing pattern using tiles. Fill in the chart for steps 4-8. Graph the pattern using the step number and the number of the new tiles. Graph the pattern using the step number and the total tiles. Use words to describe the rule for the number of new tiles and the rule for the total tiles. Synthesis/Evaluation Level: � Plan a story about a walking trip. Then, write the story and make a graph to go with the

story using graph paper. Explain how the graph shows the changes of speed that happen in the story. Make the story, graph, and explanation clear enough to be used to teach a fourth grade student how to read the graph. You may want to add a table for the graph to make the description of the trip clearer.

Geometric patterns

numerical patterns

Ordered pair

Table of values

Supporting SOL: 5.17 The student will describe the relationship found in a number pattern and express the relationship.

Patterns and Algebra Grade 5 Mathematics Enduring Understanding 10: Change, in various contexts, both quantitative and qualitative, can be identified and analyzed.

Change and Interactions Discipline Level Concept: Cause and Effect

Input Output ( x) ( y) 15 6 11 2 18 9

Essential Standards in grade band 3-5 Use patterns in mathematical situations to investigate and describe how change in one variable relates to change in a second variable.

Performance Indicators Assessment Samples – SOL/Blooms Vocabulary All students will: � understand an equation

can be used to represent a relationship between numbers.

� understand both sides of

equal. � understand there is as

structure to how a pattern grows and changes.

� understand that mathematical relationships exist in patterns.

� be able to describe unknown quantities using letters.

� understand that numeric and geometric patterns can be expressed in words and symbols, graphs, and tables.

Knowledge/Comprehension Level: A number machine uses a rule to change numbers into different numbers. The

table below shows what happened when different numbers went into the same number machine.

Which of the following could be the rule used by this number machine?

A Divide by 2

B Subtract 9

C Multiply by 2

D Add

Application/Analysis Level: Each shape stands for a different number. All shapes that are the same stand for the same number. Find the number for the square.

+ = 35 + = 27

+ = 18 = ? Synthesis/Evaluation Level:

Equation

Equal

Numerical patterns

geometric patterns

variable

variable expression

open sentence

an equation must be

Patterns and Algebra Grade 5 Mathematics Enduring Understanding 10: Change, in various contexts, both quantitative and qualitative, can be identified and analyzed.

Change and Interactions Discipline Level Concept: Cause and Effect

John saw the beginnings of a chart on a table at school, but something spilled on it.

1 3 2 6 3 9 4 12

15 24

10 Write a situation that this could be a chart of. Find a rule that fits the chart and any future piece of data to be put on the chart. Justify your answer with examples.

SOL: 5.17 The student will describe the relationship found in a number pattern and express the relationship.

Supporting SOL: 5.18 The student will a) investigate and describe the concept of variable; b) write an open sentence to represent a given mathematical relationship, using a variable; c) model one-step linear equations in one variable, using addition and subtraction; and d) create a problem situation based on a given open sentence, using a single variable.

Number and Operations Grade 5 Mathematics Enduring … · 2018-03-30 · Number and Operations...

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