Download 12 | Page 28 Determine whether mathematical

28
Determine whether
mathematical statements of
equality and inequalities are
true or false.
Write three different equations that have
a solution of 10. Write each equation and
prove it is accurate by solving for the
variable.
Replace variables with values
that make equations and
statements of inequality true.
29
Solve one-step addition and
subtraction equations using
bar models. Solve one-step
equations by using inverse
operations to isolate and solve
for the variable.
30
Use substitution to check the
accuracy of your work.
31
Solve one-step multiplication
and division equations using
bar models. Solve one-step
equations by using inverse
operations to isolate and solve
for the variable.
32
Engage NY
Module 4
Lessons 23 -24
(Appendix C)
My Math
Chapter 7 Lesson 1
•
•
Pacing – 2 days
Begin with bar models as a concrete
representation (engage NY), then
connect this to the concept of using
inverse operations to isolate and
solve for the variable (My Math)
Draw a bar diagram then write and solve
an equation to solve the following word
problem:
Patsy and LaShonda together have saved
up $553.25 to buy a TV that costs
$725.34.
Engage NY
Module 4
Lesson 26
(Appendix C)
My Math
Chapter 7 Lessons 2-3
Write and solve an equation to solve for
how much more money they need to
raise.
•
•
Pacing – 2 days
Begin with bar models as a concrete
representation (engage NY), then
connect this to the concept of using
inverse operations to isolate and
solve for the variable (My Math)
Use tape diagrams to find the solution of
!
= 4.
!"
Engage NY
Module 4
Lesson 27
(Appendix C)
My Math
Chapter 7 Lessons 4-5
Use substitution to check the
accuracy of your work.
12 | P a g e 33
Represent and solve real
world problems using onestep equations
In a given amount of time, Jamie drove
twice as far as Rhonda. Altogether they
drove 90 miles. Write an equation to find
the number of miles driven by each.
Use substitution and tape
diagrams to check the
accuracy of your work
34
Apply knowledge of
simplifying expressions, order
of operations and properties
of equality to calculate the
solution of multi-step
equations.
35
13 | P a g e Engage NY
Module 4 Lesson 28
(Appendix C)
https://www.illustrative
mathematics.org/illustr
ations/1107
A pet store owner, Byron, needs to
determine how much food he needs to
feed the animals. Byron knows that he
needs to order the same amount of bird
food as hamster food. He needs four
times as much dog food as bird food and
needs half the amount of cat food as dog
food. If Byron orders 600 packages of
animal food, how much dog food does he
buy? Let 𝑏 represent the number of
packages of bird food Byron purchased
for the pet store. Engage NY
Module 4 Lesson 29
(Appendix C)
Flex Day (Instruction Based on Data)
Recommended Resources:
“Bake Sale Brownies” (Appendix C)
My Math Chapter 7 21st Century Careers (Pages 569 – 570)
My Math Chapter 7 Review (Pages 571 – 574)
http://www.ixl.com/math/grade-6/does-x-satisfy-the-equation
http://www.ixl.com/math/grade-6/solve-one-step-equations-with-whole-numbers
http://www.ixl.com/math/grade-6/solve-word-problems-involving-two-variable-equations
http://www.opusmath.com/common-core-standards/6.ee.7-solve-real-world-and-mathematical-problems-by-writing-and-solving
36
Given an equation with two
variables, determine which is
the independent variable and
which is the dependent
variable
•
Determine whether or not the
equation is solved for the
second variable in terms of
the first variable or vice versa
•
37
Graph one-step equations on
the coordinate plane
•
•
•
14 | P a g e create a table by placing the
independent variable in the first row
or column and the dependent
variable in the second row or
column. They compute entries in
the table by choosing arbitrary
values for the independent variable
(no constraints) and then determine
what the dependent variable must
be.
Students understand that the
dependent variable changes
proportionally to the changes in the
independent variable.
What is the relationship between the two
variables shown in the table? Write an
equation that illustrates the relationship.
Explain your reasoning.
On the four square do now, assess
students’ ability to plot points in the
first quadrant of the coordinate plane
Students understand that a set of
ordered pairs make the equation true
and that the y value, or dependent
variable, is the result of substituting
numbers for the x value, or
independent variable.
Students understand that the
relationship between two variables
may be represented on a graph
where the independent variable is on
the x axis and the dependent variable
is on the y axis and analyzed in
relationship to the equation.
Write about a real-world situation that
can represented by the equation y = 1/2x.
Explain what variables represent in the
situation. Solve your scenario and graph
it below.
Engage NY
Module 4 Lesson 31
(Appendix C)
My Math
Chapter 8 Lesson 1
https://www.khanaca
demy.org/math/cc-­‐
sixth-­‐grade-­‐math/cc-­‐
6th-­‐expressions-­‐and-­‐
variables/cc-­‐6th-­‐
dependent-­‐
independent/v/depen
dent-­‐and-­‐
independent-­‐
variables-­‐exercise-­‐
example-­‐1 Engage NY
Module 4 Lesson 32
(Appendix C)
My Math Chapter 8 Lesson 3 https://www.khanacade
my.org/math/cc-sixthgrade-math/cc-6thexpressions-andvariables/cc-6thdependentindependent/v/depende
nt-and-independentvariables-exerciseexample-2
38
39
Represent graphs with an
equation
Students recognize
independent and dependent
variables in a real world
context and represent the
relationship in various ways
(graphs, equations, tables)
Use the graph below to answer the
question:
• Pacing – 2 days
• Additional Practice:
Stephanie is helping her band collect
money to fund a field trip. The band
decided to sell boxes of chocolate bars.
Each bar sells for $1.50 and each box
contains 20 bars. Below is a partial table
of monies collected for different
numbers of boxes sold.
40
(a) Complete the table above
15 | P a g e Circle the equation that the graph above
matches:
Y = 4 – 6x
Y = 6 - (1/4)x
Explain how you know:
A table can be used to show the
relationship between the number of hours
a painter works painting and the total
amount the painter charges for painting.
The painter charges $25 per hour to paint
a room.
“Making Sense of Graphs” (Appendix C) https://www.khanacade
my.org/math/cc-sixthgrade-math/cc-6thexpressions-andvariables/cc-6thdependentindependent/v/depende
nt-and-independentvariables-exerciseexample-3
My Math Chapter 8 Lesson 4 “Making Sense of Tables” (Appendix C) http://ccsstoolbox.agile
mind.com/parcc/about_
middle_3788.html
(a) Complete the table to show the
relationship between h, the number of
hours the painter works, and c, the total
http://www.opusmat
amount, in dollars, the painter charges for
h.com/common-­‐core-­‐
painting.
standards/6.ee.9-­‐use-­‐
(b) Write an equation that can be used to variables-­‐to-­‐
represent-­‐two-­‐
find c, the total charge for h hours of
quantities-­‐in-­‐a-­‐real-­‐
painting.
world-­‐problem-­‐that (b) Write an equation for the amount of
money, m, that will be collected
if b boxes of chocolate bars are sold.
Which is the independent variable and
which is the dependent variable?
(c) Calculate how much money will be
collected if 100 boxes of chocolate bars
are sold.
(d) The band collected $1530.00 from
chocolate bar sales. How many boxes
did they sell?
It took the painter 13 hours to paint a
room.
(c) What is the total amount, in dollars,
the painter charged for painting the
room? Show or explain how you got your
answer. http://www.yummy
math.com/tag/6-­‐ee-­‐
9/ 41
Flex Day (Instruction Based on Data)
Recommended Resources:
“Who Has Faulty Thinking?” (Appendix C)
My Math Chapter 8 Lesson 2
My Math Chapter 8 Problem-Solving Investigation (Pages 611 – 613)
My Math Chapter 8 Mid-Chapter Check (Page 614)
My Math Chapter 8 Inquiry Lab (Pages 615 – 616)
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&sqi=2&ved=0CDIQFjAC&url=http%3A%2F%2Fwww.yu
mmymath.com%2Ftag%2F6-ee-9%2F&ei=GWzqUsWFFPfJsQTA0oCICQ&usg=AFQjCNHDjiwJfa3UJugY1e8u8DIP23sEA&bvm=bv.60444564,d.aWc
42
Determine whether a given
value is a solution to an
inequality. Find possible
solutions to inequalities
•
•
16 | P a g e Work with your student to
distinguish between inequalities and
equations. Ask your student to solve
an equation like 6y - 3 = 15. Once
they have found the solution, ask
them to substitute the = for a > sign
and ask them to solve. Discuss how
the answer has changed
Check out this LearnZillion video
for additional ideas on how to teach
this concept:
http://learnzillion.com/lessons/3774.
Twelve is less than 3 times another
number can be shown by the inequality
12 < 3n What numbers could possibly
make this a true statement?
Engage NY
Module 4 Lesson 33
(Appendix C)
My Math
Chapter 8 Lesson 5
http://www.opusmath.c
om/common-corestandards/6.ee.5understand-solving-anequation-or-inequalityas-a-process-ofanswering-a
43
Determine when a real world
situation should be
represented by an inequality
rather than an equation.
•
Graph the range of solutions.
Tom wants to buy a pair of shoes and a tshirt. The shoes cost $65. He has $82 to
spend.
“When is it Not
Equal?”
(Appendix C)
1. Write an inequality to represent this
situation.
“Evaluating Solutions
to One-Step Equations
and Inequalities”
(Appendix C)
2. What could the cost of the t-shirt be?
Explain how you know using words.
44
Write and graph the range of
solutions to an inequality.
Explain why there are an
•
infinite number of solutions to
the inequality x > c or x <
•
17 | P a g e Key Understandings:
1) Students recognize that
inequalities of the form x < c and
x > c, where x is a variable and
c is a fixed number have
infinitely many solutions when
the values of x come from a set
of rational numbers.
Work with your students to
recognize the values on a number
line that will satisfy an inequality by
graphing them and then inserting
them and determining if the value
satisfies the inequality. For
additional ideas, use this
LearnZillion lesson:
http://learnzillion.com/lessons/2775graph-an-inequality
Little Ceasar’s charges $5.00 for a pizza.
Kervin will spend no more than $40.00.
How many pizzas can he buy? Write an
inequality to solve and graph on a
number line.
My Math
Chapter 8 Lesson 6
Engage NY
Module 4 Lesson 34
(Appendix C)
My Math
Chapter 8 Lesson 7
http://www.opusmath.c
om/common-corestandards/6.ee.8-writean-inequality-of-theform-x--c-or-x--c-torepresent-a-constraint
45
Determine the range of
solutions that would make an
inequality true in a real world
problem
(2 days)
46
18 | P a g e Use these task from Illustrative
Mathematics to create task-based
problem-solving lessons:
Task I:
Fishing Adventures rents small fishing
boats to tourists for day-long fishing
trips. Each boat can hold at most eight
people. Additionally, each boat can only
carry at most 900 pounds of weight for
safety reasons.
(a) Let p represent the total number of
people on a boat. Write an inequality to
describe the number of people that a
boat can hold. Draw a number line
diagram that shows all possible
solutions to the inequality.
(b) Let w represent the total weight of a
group of people wanting to rent a boat.
Write an inequality that describes all
total weights allowed in a single boat.
Draw a number line diagram that shows
all possible solutions to the inequality.
Task II:
A theme park has a log ride that can
hold 12 people. They also have a weight
limit of 1500 pounds for safety reasons.
If the average adult weighs 150 pounds,
the average child weighs 100 pounds,
and the log itself weighs 200 pounds,
the ride can operate safely if the
following inequality is satisfied:
150A + 100C + 200 ≤ 1500.
Part I: Ana is saving to buy a bicycle that
costs $135. She has saved $98 and wants
to know how much more money she
needs to buy the bicycle. The equation
135 = x + 98 models this situation,
where x represents the additional amount
of money Ana needs to buy the bicycle.
https://www.illustrative
mathematics.org/illustr
ations/673
http://www.opusmat
h.com/common-­‐core-­‐
standards/6.ee.8-­‐
(a) When substituting for x, which
write-­‐an-­‐inequality-­‐
value(s), if any, from the set {0, 37, 98,
of-­‐the-­‐form-­‐x-­‐-­‐c-­‐or-­‐x-­‐
135, 233} will make the equation true?
-­‐c-­‐to-­‐represent-­‐a-­‐
constraint (b) Explain what this means in terms of
the amount of money needed and the cost http://www.internet4
of the bicycle.
classrooms.com/com
mon_core/write_ineq
Part II: Ana considered buying the $135
uality_form_x_c_expre
bicycle, but then she decided to shop for ssions_equations_sixt
a different bicycle. She knows the other
h_6th_grade_math_ma
bicycle she likes will cost more than
thematics.htm $150. This situation can be modeled by
the inequality x + 98 > 150.
(c) Which values of x, if any, from −250
to 250 will make the inequality true? If
more than one value makes the inequality
true, identify the least and greatest values
that make the inequality true.
(d) Explain what this means in terms of
the amount of money needed and the cost
of the bicycle.
A is the number of adults and C is the
number of children on the log ride.
There are several groups of different
numbers of children waiting to ride.
Group A has 4 children, group B has 3
children, group C has 9 children, group
D has 6 children, and group E has 5
children.
If 4 adults are already seated in the
log, which groups of children can
safely ride with them? Which cannot?
Explain your answers.
47
Flex Day (Instruction Based on Data)
Recommended Resources:
“Want Ads” (Appendix C)
My Math Chapter 8 21st Century Career (Pages 643 – 644)
My Math Chapter 8 Review (Pages 645 – 648)
My Math Unit Project (Pages 649 – 650)
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&sqi=2&ved=0CDIQFjAC&url=http%3A%2F%2Fwww.yu
mmymath.com%2Ftag%2F6-ee-9%2F&ei=GWzqUsWFFPfJsQTA0oCICQ&usg=AFQjCNHDjiwJfa3UJugY1e8u8DIP23sEA&bvm=bv.60444564,d.aWc
48
MCLASS End of Unit Assessment
Appendix B
*Note: This assessment will be administered online
19 | P a g e Appendix A:
Unpacked Standards Guide
Source: Public Schools of North Carolina NCDPI Collaborative Workspace Unpacking
What do these standards mean a child will know and be able to do?
Standard
6.EE.1 Write and evaluate numerical
expressions involving whole-number
exponents.
€
6.EE.1 Students demonstrate the meaning of exponents to write and evaluate numerical expressions with whole number
1
1 1 1
exponents. The base can be a whole number, positive decimal or a positive fraction (i.e. 5 can be written • • •
2
2 2 2
1 1
1
• which has the same value as
). Students recognize that an expression with a variable represents the same
2 2
32
mathematics (ie. x5 can be written as x • x • x • x • x) and write algebraic expressions from verbal expressions.
€
€ € €
€
Order of operations is introduced throughout elementary grades, including the use of grouping symbols, ( ), { }, and [ ]
€
in 5th grade. Order of operations
with exponents is the focus in 6th grade.
Example1:
What is the value of:
•
0.23
Solution: 0.008
•
5 + 24 • 6
Solution: 101
•
72 – 24 ÷3 + 26
Solution: 67
Example 2:
What is the area of a square with a side length of 3x?
Solution: 3x • 3x = 9x2
6.EE.2 Write, read, and evaluate
expressions in which letters stand for
numbers.
a. Write expressions that record
operations with numbers and
with letters standing for numbers.
For example, express the
20 | P a g e Example 3:
4x = 64
Solution: x = 3 because 4 • 4 • 4 = 64
6.EE.2 Students write expressions from verbal descriptions using letters and numbers, understanding order is important
in writing subtraction and division problems. Students understand that the expression “5 times any number, n” could be
represented with 5n and that a number and letter written together means to multiply. All rational numbers may be used
in writing expressions when operations are not expected. Students use appropriate mathematical language to write
verbal expressions from algebraic expressions. It is important for students to read algebraic expressions in a manner
that reinforces that the variable represents a number.
calculation “Subtract y from 5”
as 5 – y.
Example Set 1:
Students read algebraic expressions:
• r + 21 as “some number plus 21” as well as “r plus 21”
• n • 6 as “some number times 6” as well as “n times 6”
•
s
6 and s ÷ 6 as “as some number divided by 6” as well as “s divided by 6”
Example Set 2:
Students write algebraic expressions:
€ • 7 less than 3 times a number
Solution: 3x – 7
• 3 times the sum of a number and 5
Solution: 3 (x + 5)
• 7 less than the product of 2 and a number
Solution: 2x – 7
• Twice the difference between a number and 5
Solution: 2(z – 5)
• The quotient of the sum of x plus 4 and 2
Solution: x + 4
2
Students can describe expressions such as 3 (2 + 6) as the product of two factors: 3 and (2 + 6). The quantity
(2 + 6) is viewed as one factor consisting of two terms.
Terms are the parts of a sum. When the term is an explicit number, it is called a constant. When the term is a product of
a number and a variable, the number is called the coefficient of the variable.
Students should identify the parts of an algebraic expression including variables, coefficients, constants, and the names
of operations (sum, difference, product, and quotient). Variables are letters that represent numbers. There are various
possibilities for the number they can represent.
Consider the following expression:
x2 + 5y + 3x + 6
The variables are x and y.
There are 4 terms, x2, 5y, 3x, and 6.
There are 3 variable terms, x2, 5y, 3x. They have coefficients of 1, 5, and 3 respectively. The coefficient of x2 is 1,
since x2 = 1x2. The term 5y represent 5y’s or 5 • y.
There is one constant term, 6.
The expression represents a sum of all four terms.
21 | P a g e b. Identify parts of an expression
using mathematical terms (sum,
term, product, factor, quotient,
coefficient); view one or more
parts of an expression as a single
entity. For example, describe the
expression 2 (8 + 7) as a product
of two factors; view (8 + 7) as
both a single entity and a sum of
two terms.
c. Evaluate expressions at specific
values of their variables. Include
expressions that arise from
formulas used in real-world
problems. Perform arithmetic
operations, including those
involving whole- number
exponents, in the conventional
order when there are no
parentheses to specify a
particular order (Order of
Operations). For example, use
the formulas V = s3 and A = 6 s2
to find the volume and surface
area of a cube with sides of length
s = ½.
Students evaluate algebraic expressions, using order of operations as needed. Problems such as example 1 below require
students to understand that multiplication is understood when numbers and variables are written together and to use the
order of operations to evaluate.
Order of operations is introduced throughout elementary grades, including the use of grouping symbols, ( ), { }, and [ ]
in 5th grade. Order of operations with exponents is the focus in 6th grade.
Example 1:
Evaluate the expression 3x + 2y when x is equal to 4 and y is equal to 2.4.
Solution:
3 • 4 + 2 • 2.4
12 + 4.8
16.8
Example 2:
1
Evaluate 5(n + 3) – 7n, when n = 2 .
Solution:
1
1
5( 2 + 3) – 7( 2 )
€
1
1
1
7
1
5 (3 2 ) - 3 2
Note: 7( 2 ) = 2 = 3 2
€
€
1
1
1
1
17
€
€ 2 -32
€ Students
€
€may also reason that 5 groups of 3 2 take away 1 group of 3 2 would give 4
1
1
14
groups of 3 2 . Multiply 4 times 3 2 to get 14.
€
€
€
€
22 | P a g e €
€