Download 6-7th Grade Mathematics Curriculum Guide

6-7th Grade Mathematics Curriculum Guide
Unit 4: Expressions
Time Frame: Quarter 2 – about 20 days
2015 – 2016
Connections to Previous Learning:
Students have been writing numerical expressions since Kindergarten, such as 2 + 3. In grade 5 they used whole number exponents to express powers of 10, and in Grade 6 they
start to incorporate whole number exponents into numerical expressions Students have also been using letters to represent an unknown quantity in word problems since Grade
3. In Grade 6 they begin to work systematically with algebraic expressions. Students in Grade 5 began to move from viewing expressions as actions describing a calculation to
viewing them as objects in their own right. They describe the structure of expressions, seeing them as a product of two factors, the second of which can be viewed as both a
single entity and a sum of two terms.
Focus of the Unit:
In this unit, students understand the use of variables in mathematical expressions. They become more fluent at viewing expressions as objects in their own right versus
calculations. Students write expressions that correspond to given situations, evaluate expressions, and use expressions and formulas to solve problems.
Students will understand that expressions in different forms can be equivalent, and they will use the properties of operations to generate and rewrite expressions in equivalent
forms. The Mathematical Practices should be evident throughout instruction of symbolic expressions and connected to the content.
Students should engage in mathematical tasks that provide an opportunity to connect content and practices.
Connections to Subsequent Learning:
Students extend their previous understandings of numerical expressions. They continue to develop their understanding of how letters are used to represent numbers in
mathematics, in both equality and inequality statements. They will use tables, words, numbers, graphs, and equations to describe relationships.
Students will solve complex problems in problem solving situations using expressions including the use of the idea of dependent and independent variables.
Mathematical Practices
1. Make Sense of Problems and Persevere in Solving Them.
2. Reason Abstractly and Quantitatively.
3. Construct Viable Arguments and Critique the Reasoning of Others.
4. Model with Mathematics.
5. Use Appropriate Tools Strategically.
6. Attend to Precision.
7. Look for and Make Use of Structure.
8. Look for and Express Regularity in Repeated Reasoning.
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6-7th Grade Mathematics Curriculum Guide
Stage 1 Desired Results
2015 – 2016
Transfer Goals
Students will be able to independently use their learning to…
• Generalize a pattern.
• Use patterns to make predictions.
Meaning Goals
UNDERSTANDINGS
Students will understand that…
6th
• Properties of operations are used to determine if expressions are
equivalent.
• There is a designated sequence to perform operations (Order of
Operations).
• Variables can be used as unique unknown values or as quantities
that vary.
• Algebraic expressions may be used to represent and generalize
mathematical problems and real life situations
ESSENTIAL QUESTIONS
6th
• What is equivalence?
• How properties of operations used to prove equivalence?
• How are variables defined and used?
7th
• When and how are expressions, equations, inequalities and graphs applied to real world situations?
• How can the order of operations be applied to evaluating expressions, and solving from one-step to
multi-step equations?
7th
• Variables can be used to represent numbers in any type of
mathematical problem.
• Understand the difference between an expression and an equation.
• Expressions you simplify and equations you solve for the variable’s
value.
• Write and solve multi-step equations including all rational
numbers.
• Some equations may have more than one solution and understand
inequalities.
• Properties of operations allow us to add, subtract, factor, and
expand linear expressions.
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6-7th Grade Mathematics Curriculum Guide
Acquisition Goals
Students will know…
• The difference between an equation and expression.
• Properties are used to solve one-step equations.
• Substitution can be used to verify a solution to an equation.
• Equations represent situations.
• Order of operations.
• The distributive, commutative, associative, zero and identity
properties.
Unit 1
2015 – 2016
Students will be skilled at…
• Write numerical expressions that have whole number exponents. (6.EE.1)
• Evaluate numerical expressions that have whole number exponents and rational bases.(6.EE.1)
• Write algebraic expressions to represent real life and mathematical situations (6.EE.2)
• Identify parts of an expression using appropriate terminology (6.EE.2)
• Given the value of a variable, students will evaluate the expression (6.EE.2)
• Use order of operations to evaluate expressions. (6.EE.2)
• Apply properties of operations to write equivalent expressions. (6.EE.3)
• Identify when two expressions are equivalent. (6.EE.4)
• Prove (using various strategies) that two equations are equivalent no matter what
number is substituted. (6.EE.4)
• Identify the factors of any whole number less than or equal to 100. (6.NS.4)
• Identify the multiples of two whole numbers less than or equal to 12 and determine the Least Common
Multiple. (6.NS.4)
• Use the distributive property to express a sum of two whole numbers 1-100 with a
common factor as a multiple of a sum of two whole numbers with no common factor. (6.NS.4).
• Use Commutative, Associative, Distributive, Identity, and Inverse Properties to add and subtract linear
expressions with rational coefficients. (7.EE.1)
• Use Commutative, Associative, Distributive, Identity, and Inverse Properties to factor and expand linear
expressions with rational coefficients. (7.EE.1)
• Rewrite an expression in a different form. (7.EE.2)
• Choose the form of an expression that works best to solve a problem. (7.EE.2)
• Explain your reasoning for the choice of expression used to solve a problem. (7.EE.2)
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6-7th Grade Mathematics Curriculum Guide
2015 – 2016
Stage 1 Established Goals: Common Core State Standards for Mathematics
6.EE.A Apply previous understandings of arithmetic to algebraic expressions
6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.2 Write, read and evaluate expressions in which letters stand for numbers.
6.EE.A.2.a Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
6.EE.A.2.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
entry. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both as single entity and a sum of two terms.
6.EE.A.2.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 = 62 to find the volume and surface area of a cube with sides of lengths s =1/2.
6.EE.A.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to
produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to
y + y + y to produce the equivalent expression 3y.
6.EE.A.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into
them). For example, the expressions y + y + y and 3y are equivalent because the name the same number regardless of which number y stands for.
7.EE.A Use properties of operations to generate equivalent expressions.
7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.A.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a +0.05a
= 1.05a means that “increase by 5% is the same as multiply by 1.05.”
Explanations, Examples, and Comments
Overview of Suggested Assessments & Performance Tasks for Unit
Vocabulary for Unit 3
superscripted numbers, equivalent, base number, coefficient, exponents, power, equation,
expression, variables, order of operations, numerical expression, algebraic expression, term,
distributive property, commutative property, constant, like terms, sum, product, factor, quotient,
equivalent expressions
Mid-Unit Assessment –
End-of-Unit Assessment
Formative assessments for this unit
6.EE.A.2c
6.EE.3
7.EE.1
Suggested Performance Task for this unit
From Georgia Department Of Education: (available on P: drive)
• Exponents 6.EE.1
• Rules for Exponents 6.EE.1
• Conjectures about Properties 6.EE.3
• Writing and Evaluating Expressions 6.EE.1, 6.EE.2, 6.EE.3. 6.EE.4
• Are We Equal? 6.EE.3. 6.EE.4
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Howard County
• Shop Smart – 7.EE.2
2015 – 2016
st
1 Semester Summative (sent to building)
6.EE.A.1
Vocab - superscripted numbers, base number, exponents, power, expression, numerical
expression, algebraic expression
•
•
Write the following as a numerical expressions using exponential notation.
o The area of a square with a side length of 8 m (Solution: 8²m²
o The volume of a cube with a side length of 5 ft.: (Solution: 5³ft³ )
o Yu‐Lee has a pair of mice. The mice each have 2 babies. The babies grow up and
have two babies of their own: (Solution: 2³ mice)
Evaluate:
o 4³ (Solution: 64)
o 5 + 2⁴ * 6 (Solution: 101)
o 7² - 24 ÷ 3 + 26 (Solution: 67)
6.EE.A.2
Vocab – coefficient, variables, order of operations, term, constant, sum, product, factor, quotient
It is important for students to read algebraic expressions in a manner that reinforces that the
variable represents a number.
o r + 21 as “some number plus 21 as well as “r plus 21”
o n * 6 as “some number times 6 as well as “n times 6”
𝒔
o
and s ÷ 6 as “as some number divided by 6” as well as “s divided by 6”
Stage 3 MATERIALS BY STANDARD(S):
Teacher should use assessment data to determine which of the materials below
best meet student instructional needs. All materials listed may not be needed.
6.EE.A.1
Holt Course 2 Lesson 1-1 Numbers and Patterns (1)
Holt Course 2 Lesson 1-2 Exponents (1)
Holt Course 2 Lesson 1-4 Applying Exponents (1)
Holt Course 2 Lesson 1-5 Order of Operations (1)
Problem Solving Resources
Making Sense of Problem Solving G.4 Math Magic
6.EE.A.2
Holt Course 1 Lesson 2-3 Translating Between Tables and Expressions (1)
Holt Course 2 Lesson 1-7 Variables and Algebraic Expressions (1)
Holt Course 2 Lesson 1-8 Translate Words into Math (1)
Engage NY Module 4 Lesson 15-17 (2)
http://www.engageny.org/resource/grade-6-mathematics-module-4
𝟔
Students should identify the parts of an algebraic expression including variables, coefficients,
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2015 – 2016
constants, and the names of operations (sum, difference, product, and quotient). Development
of this common language helps students to understand the structure of expressions and explain
their process for simplifying expressions.
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.
Variables are letters that represent numbers. There are various possibilities for the numbers they
can represent; students can substitute these possible numbers for the letters in the expression
for various different purposes.
Consider the following expression:
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 5 y’s or 5 * y.
There is one constant term, 6.
The expression shows a sum of all four terms.
Examples:
• 7 more 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) )
1
• Evaluate 5(n + 3) – 7n, when n =
2
•
The expression c + 0.07c can be used to find the total cost of an item with 7% sales tax,
where c is the pre-tax cost of the item. Use the expression to find the total cost of an item
that cost $25.
The perimeter of a parallelogram is found using the formula p = 2l + 2w. What is the perimeter of
a rectangular picture frame with dimensions of 8.5 inches by 11 inches.
6.EE.A.3
Vocab – equivalent, coefficient, distributive property, equivalent expressions
Students use their understanding of multiplication to interpret 3 (2 + x). For example,
Unit 1
6.EE.A.3
Holt Course 2 Lesson 1-6 Properties (1)
Engage NY Module 4 Lesson 11 & 12 (2)
http://www.engageny.org/resource/grade-6-mathematics-module-4
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3 groups of (2 + x). They use a model to represent x, and make an array to show the meaning of
3(2 + x). They can explain why it makes sense that 3(2 + x) is equal to 6 + 3x.
An array with 3 columns and x + 2 in each column:
Students interpret y as referring to one y. Thus, they can reason that one y plus one y plus one y
must be 3y. They also the distributive property, the multiplicative identity property of 1, and the
commutative property for multiplication to prove that
y + y + y = 3y:
y + y + y = y x 1 + y x 1 + y x 1 = y x (1 + 1 + 1) = y x 3 = 3y
6.EE.4/7.EE.A.1
6.EE.A.4/7.EE.A.1
Holt Common Core Companion Course 1 Equivalent Expressions Supplement
Lesson 4-3A TE (p. 96) & SE (P. 8) (1) EXTRAS
Vocab – coefficient, commutative property, like terms, factor
6.EE.A.4
Students connect their experiences with finding and identifying equivalent forms of whole
numbers and can write expressions in various forms. Students generate equivalent expressions
using the associative, commutative, and distributive properties. They can prove that the
expressions are equivalent by simplifying each expression into the same form.
Example:
•
Are the expressions equivalent? How do you know?
4m + 8
4(m+2)
3m + 8 + m
2 + 2m + m + 6 + m
7.EE.A.1
Engage NY Module 3 Lesson 1 & 2 (1)
http://www.engageny.org/resource/grade-7-mathematics-module-3
Holt Course 3 Lesson 11-1 Simplifying Algebraic Expressions (1)
Additional Resources, if needed
Holt Course 2 Lesson 1-9 Simplifying Algebraic Expressions
Solution:
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7.EE.1
This is a continuation of work from 6th grade using properties of operations (table 3, pg. 90) and
combining like terms. Students apply properties of operations and work with rational numbers
(integers and positive / negative fractions and decimals) to write equivalent expressions.
Examples:
• Write an equivalent expression for 3(x + 5) - 2.
• Suzanne thinks the two expressions 2(3a – 2) + 4a and 10a - 2is equivalent? Is she correct?
Explain why or why not?
•
Write equivalent expressions for: 3a + 12.
Possible solutions might include factoring as in 3 (a + 4), or other expressions such as a + 2a +7
+7.
•
A rectangle is twice as long as wide. One way to write an expression to find the perimeter
would be w + w + 2w + 2w . Write the expression in two other ways.
Solution: 6w OR 2(w)+ 2(2w).
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•
An equilateral triangle has a perimeter of 6x + 15. What is the length of each of the sides of
the triangle?
Solution: 3(2x + 5), therefore each side is 2x + 5 units long.
7.EE.A.2
Students understand the reason for rewriting an expression in terms of a contextual situation.
For example, students understand that a 20% discount is the same as finding 80% of the cost, c
(0.80c).
Example 1:
All varieties of a certain brand of cookies are $3.50. A person buys peanut butter cookies and
chocolate chip cookies. Write an expression that represents the total cost, T, of the cookies if p
represents the number of peanut butter cookies and c represents the number of chocolate chip
cookies
7.EE.A.2
Engage NY Module 3 Lesson 3-5 (2)
http://www.engageny.org/resource/grade-7-mathematics-module-3
Suggested Performance Task
Howard County
• Shop Smart – 7.EE.2
Solution:
Students could find the cost of each variety of cookies and then add to find the total.
T = 3.50p + 3.50c
Or students could recognize that multiplying 3.50 by the total number of boxes (regardless of
variety) will give the same total.
T = 3.50(p +c)
Example 2:
Jamie and Ted both get paid an equal hourly wage of $9 per hour. This week, Ted made an
additional $27 dollars in overtime. Write an expression that represents the weekly wages of both
if J = the number of hours that Jamie worked this week and T = the number of hours Ted worked
this week? What is another way to write the expression?
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Solution:
Students may create several different expressions depending upon how they group the
quantities in the problem.
Possible student responses are:
2015 – 2016
Response 1: To find the total wage, first multiply the number of hours Jamie worked by 9. Then,
multiply the number of hours Ted worked by 9. Add these two values with the $27 overtime to
find the total wages for the week. The student would write the expression 9J + 9T + 27.
Response 2: To find the total wages, add the number of hours that Ted and Jamie worked. Then,
multiply the total number of hours worked by 9. Add the overtime to that value to get the total
wages for the week. The student would write the expression 9(J + T) + 27.
Response 3: To find the total wages, find out how much Jamie made and add that to how much
Ted made for the week. To figure out Jamie’s wages, multiply the number of hours she worked
by 9. To figure out Ted’s wages, multiply the number of hours he worked by 9 and then add the
$27 he earned in overtime. My final step would be to add Jamie and Ted wages for the week to
find their combined total wages. The student would write the expression (9J) + (9T + 27).
Example 3:
Given a square pool as shown in the picture, write four different expressions to find the total
number of tiles in the border. Explain how each of the expressions relates to the diagram and
demonstrate that the expressions are equivalent. Which expression is most useful? Explain.
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Evaluative Criteria/Assessment Level Descriptors (ALDs):
6.EE.A (SBAC Target E)
Level 4 students should be able to apply the understanding of the
properties of operations and use the properties to show why two
expressions are equivalent.
Level 3 students should be able to write and evaluate numerical
expressions with nonnegative integer exponents and expressions from
formulas in real-world problems, and they should be able to apply and
extend previous understandings of arithmetic to evaluate expressions with
variables that include nonnegative integer exponents. They should be able
to apply properties of operations to generate equivalent expressions.
Level 2 students should be able to evaluate numerical expressions with
nonnegative integer exponents that do not need to be distributed across a
set of parentheses. They should be able to apply and extend previous
understandings of arithmetic to evaluate expressions with variables that do
not contain exponents. They should also be able to write one- and two-step
algebraic expressions that introduce a variable and identify equivalent
expressions.
Level 1 students should be able to evaluate numerical expressions without
exponents; write one- or two-step numerical expressions; and identify parts
of an expression, using terms (e.g., coefficient, term, sum, product,
difference, quotient, factor).
2015 – 2016
Stage 2 - Evidence
OTHER ASSESSMENT EVIDENCE:
Formative assessments (available on P: drive)
Spaced Learning Over Time (SLOT - entrance & exit slips)
Checking for understanding (CFU’s)
Lesson Quizzes
Classroom Assessments
7.EE.A (SBAC Target C)
Level 4 students should understand that rewriting an expression can shed
light on how quantities are related in an unfamiliar problem-solving context
with no scaffolding.
Level 3 students should be able to apply properties of operations as
strategies to factor and expand linear expressions with rational coefficients.
They should understand that rewriting an expression can shed light on how
quantities are related in a familiar problem-solving context with minimal
scaffolding.
Level 2 students should be able to apply properties of operations as
strategies to factor and expand linear expressions with integer coefficients.
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6-7th Grade Mathematics Curriculum Guide
They should also be able to add and subtract linear expressions with
rational coefficients.
Level 1 students should be able to apply properties of operations as
strategies to add and subtract linear expressions with integer coefficients.
LEARNING ACTIVITIES:
2015 – 2016
Stage 3 – Learning Plan Sample
Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
NOTES:
**Days may change depending on any tasks or assessing you choose to do.
6.EE.A.1
Optional: Review if needed, Exponents
• Holt Course 2 Lesson 1-2
Day 1: Order of Operations – Remember GEMA
• Holt Course 2 Lesson 1-5
6.EE.A.2
Day 2: Translating Between Tables and Expressions
• Holt Course 1 Lesson 2-3
Day 3: Variables and Algebraic Expressions
• Holt Course 2 Lesson 1-7
Day 4: Translate Words into Math
• Holt Course 2 Lesson 1-8
Day 5: Write Expressions in Which Letters Stand for Numbers
th
• Engage NY 6 Module 4Lesson 16 & 17
•
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Stage 3 – Learning Plan Sample
6.EE.A.3, 6.EE.A.4/7.EE.1
2015 – 2016
Day 6: Factoring Expressions
th
• Engage NY 6 Module 4 Lesson 11
Day 7: Distributive Property
• Engage NY Lesson 12
Day 8: Equivalent Expressions
• Holt Common Core Course 1 Supplement Lesson 4-3A
Day 9& 10: Generating Equivalent Expressions
th
• Engage NY 7 Module 3 Lesson 1 & 2
Day 11: Simplifying Algebraic Expressions
• Holt Course 3 Lesson 11-1
7.EE.A.2
Day 12 & 13: Writing Products as Sums and Sums as Products
th
• Engage NY 7 Module 3 Lesson 3 & 4
Day 14: Using the Identity and Inverse to Write Equivalent Expressions
th
• Engage NY 7 Module 3 Lesson 5
Daily Lesson Plan
Learning Target:
Warm-up:
Activities:
• Whole Group:
• Small Group/Guided/Collaborative/Independent:
• Whole Group:
Checking for Understanding (before, during and after):
Assessments:
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