6th Grade Mathematics
Instructional Week 21
Represent real-world problems using multiple representations and solve
equations
Paced Standards:
6.AF.5: Solve equations of the form x + p = q, x - p = q, px = q, and x/p = q fluently for cases in which p,
q and x are all nonnegative rational numbers. Represent real world problems using equations of these
forms and solve such problems.  +
6.AF.10: Use variables to represent two quantities in a proportional relationship in a real-world
problem; write an equation to express one quantity, the dependent variable, in terms of the other
quantity, the independent variable. Analyze the relationship between the dependent and
independent variables using graphs and tables, and relate these to the equation.  +
Connections to other 6th Grade Standards:
6.AF.1: Evaluate expressions for specific values of their variables, including expressions with wholenumber exponents and those that arise from formulas used in real-world problems.
6.C.4: Compute quotients of positive fractions and solve real-world problems involving division of
fractions by fractions. Use a visual fraction model and/or equation to represent these calculations.
6.AF.4: Understand that solving an equation or inequality is the process of answering the following
question: Which values from a specified set, if any, make the equation or inequality true? Use
substitution to determine whether a given number in a specified set makes an equation or inequality
true.
6.AF.8: Solve real-world and other mathematical problems by graphing points with rational number
coordinates on a coordinate plane. Include the use of coordinates and absolute value to find
distances between points with the same first coordinate or the same second coordinate.
6.AF.9: Make tables of equivalent ratios relating quantities with whole-number measurements, find
missing values in the tables, and plot the pairs of values on the coordinate plane.
6.C.6: Apply the order of operations and properties of operations (identity, inverse, commutative
properties of addition and multiplication, associative properties of addition and multiplication, and
distributive property) to evaluate numerical expressions with nonnegative rational numbers, including
those using grouping symbols, such as parentheses, and involving whole number exponents. Justify
each step in the process.
Prerequisite/Foundational Standards:
5.AT.7: Represent real-world problems and equations by graphing ordered pairs in the first quadrant
of the coordinate plane, and interpret coordinate values of points in the context of the situation.
5.AT.8: Define and use up to two variables to write linear expressions that arise from real-world
problems, and evaluate them for given values.
4.AT.6: Understand that an equation, such as y = 3x + 5, is a rule to describe a relationship between
two variables and can be used to find a second number when a first number is given. Generate a
number pattern that follows a given rule.
6th Grade ISTEP+ Toolkit
Indianapolis Public Schools
Curriculum and Instruction
6.AF.5
Standard: 6.AF.5: Solve equations of the form x + p = q, x - p = q, px = q, and x/p=q fluently for cases in
which p, q and x are all nonnegative rational numbers. Represent real world problems using equations of these forms
and solve such problems.
Teacher Background Information:
Students have used algebraic expressions to generate answers given values for the variable. This understanding is now
expanded to equations where the value of the variable is unknown but the outcome is known. For example, in the
expression, x + 4, any value can be substituted for the x to generate a numerical answer; however, in the equation x + 4 =
6, there is only one value that can be used to get a 6. Problems should be in context when possible and use only one
variable.
Students write equations from real-world problems and then use inverse operations to solve one-step equations based
on real world situations. Equations may include fractions and decimals with non-negative solutions.
1
x
1
Students recognize that dividing by 6 and multiplying by produces the same result. For example, = 9 and x = 9 will
6
6
6
produce the same result.
Beginning experiences in solving equations require students to understand the meaning of the equation and the solution
in the context of the problem.
Example 1:
Meagan spent $56.58 on three pairs of jeans. If each pair of jeans costs the same amount, write an algebraic equation
that represents this situation and solve to determine how much one pair of jeans cost.
Sample Solution:
Students might say: “I created the bar model to show the cost of the three pairs of jeans. Each bar labeled J is the same
size because each pair of jeans costs the same amount of money. The bar model represents the equation 3J = $56.58. To
solve the problem, I need to divide the total cost of $56.58 among the three pairs of jeans. I know that it will be more
than $10 each because 10 x 3 is only 30 but less than $20 each because 20 x 3 is 60. If I start with $15 each, I am up to
$45. I have $11.58 left. I then give each pair of jeans $3. That’s $9 more dollars. I only have $2.58 left. I continue until all
the money is divided. I ended up giving each pair of jeans another $0.86. Each pair of jeans costs $18.86 (15+3+0.86). I
double check that the jeans cost $18.86 each because $18.86 x 3 is $56.58.”
Example 2:
Julie gets paid $20 for babysitting. She spends $1.99 on a package of trading cards and $6.50 on lunch. Write and solve
an equation to show how much money Julie has left.
1.99
6.50
money left over (m)
Solution: 20 = 1.99 + 6.50 + m, m = $11.51
Process Standards to Emphasize with Instruction of 6.AF.5:
6.PS.1: Make sense of problems and persevere in solving them.
6.PS.2: Reason abstractly and quantitatively.
6.PS.4: Model with mathematics.
6.PS.7: Look for and make use of structure.
Indianapolis Public Schools
Curriculum and Instruction
6.AF.10
Standard: 6.AF.10 Use variables to represent two quantities in a proportional relationship in a real-world problem;
write an equation to express one quantity, the dependent variable, in terms of the other quantity, the independent
variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate
these to the equation.
Teacher Background Information:
The purpose of this standard is for students to understand the relationship between two variables, which begins
with the distinction between dependent and independent variables. The independent variable is the variable that
can be changed; the dependent variable is the variable that is affected by the change in the independent variable.
Students recognize that the independent variable is graphed on the x-axis; the dependent variable is graphed on the
y-axis.
Students recognize that not all data should be graphed with a line. Data that is discrete would be graphed with
coordinates only. Discrete data is data that would not be represented with fractional parts such as people, tents,
DVDs, etc. For example, a graph illustrating the cost per person would be graphed with points since part of a
person would not be considered. A line is drawn when both variables could be represented with fractional parts.
Discrete
Continuous
Students are expected to recognize and explain the impact on the dependent variable when the independent
variable changes (As the x variable increases, how does the y variable change?) Relationships should be
proportional with the line passing through the origin. Additionally, students should be able to write an equation
from a word problem and understand how the coefficient of the dependent variable is related to the graph and /or
a table of values.
Students can use many forms to represent relationships between quantities. Multiple representations include
describing the relationship using language, a table, an equation, or a graph. Translating between multiple
representations helps students understand that each form represents the same relationship and provides a
different perspective
Indianapolis Public Schools
Curriculum and Instruction
6.AF.10
Standard: 6.AF.10 Use variables to represent two quantities in a proportional relationship in a real-world problem;
write an equation to express one quantity, the dependent variable, in terms of the other quantity, the independent
variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and
relate these to the equation.
Teacher Background Information:
Proportional and Nonproportional Situations.
(Students will examine nonproportional relationships in 7th grade.)
Proportional
Gasoline costs $4.24 per gallon.
Two things show us that this is definitely a proportional
relationship. First, it contains the origin, (0, 0), and this
makes sense: if we buy zero gallons of gas it will cost
zero dollars. Second, if the number of gallons is doubled,
the cost is doubled; if it is tripled, the cost is tripled. The
equation that will represent this data is y = 4.24x ,
where x is the number of gallons of gasoline and y is the
total cost. The graph is shown below.
Indianapolis Public Schools
Nonproportional
The diameter of a Douglas fir tree is currently 10 inches
when measured at chest height. Over the next 50 years,
the diameter is expected to increase by an average
2
growth rate of 5 inch per year.
Two things show us that this is a nonproportional
relationship. First it contains an ordered pair (0,10)
therefore, if graphed, the line does not pass through the
origin. Second, if the number of years is doubled, the
diameter is not doubled.
2
The equation y = x+10 gives y, the diameter of the tree
5
in inches, after x years.
Curriculum and Instruction
6.AF.10
Standard: 6.AF.10 Use variables to represent two quantities in a proportional relationship in a real-world problem; write
an equation to express one quantity, the dependent variable, in terms of the other quantity, the independent variable.
Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to
the equation.
Teacher Background Information:
Functional relationships can be represented in any of five ways: (1) the pattern itself, which we can refer to as the context,
(2) the table, (3) the verbal description, (4) the symbolic equations, and (5) the graph.
Each representation is a way of looking at the function, each providing a different way of looking at or thinking about
patterns and functions, leading to deeper understanding and more flexible reasoning. Students must understand the
connections between these representations; it is not enough to just teach each one separately. That means the teacher
must pose tasks and questions that help students make these connections.
To illustrate this point, we will use the context below:
Bobby needs bananas to bake loaves of banana bread. Bobby wants to buy
bananas at his local grocery store. Bananas are sold for $2.40 per pound.
Context: The function begins with a context. Buying bananas by the pound results in a higher cost dependent upon the
number of pounds bought. The more pounds of bananas that Bobby buys the more money he will spend. The context
helps students make sense of what changes (number of pounds) and what stays the same (cost per pound). The context
helps students’ conceptual understanding of more abstract representations and illustrates that algebra is a tool for
describing real-world phenomena. The context alone is not sufficient, carefully selected tasks to connect the contexts to
other representations are needed to support students’ algebraic thinking.
Table: Bobby might sit down and calculate some possible figures based on anticipated pounds of bananas bought. This
will give him some idea of how much money he will spend depending on how many pounds of bananas he plans on
buying.
The number of pounds in the table is purely a matter of choice.
POUNDS OF BANANAS PURCHASED
One could calculate the cost for 10,000 pounds of bananas
(INDEPENDENT VARIABLE) AND COST
(10,000 x 2.4), even though it is not reasonable in this context. The
(DEPENDENT VARIABLE)
table provides a concise way to look at the recursive pattern and
Pounds of Bananas
Cost
the explicit pattern. The recursive pattern can lead to seeing what
Purchased
changes, whether it changes at a constant rate, and how that can
1
$2.40
help find the explicit equation.
2
$4.80
10
50
100
Indianapolis Public Schools
$24.00
$120.00
$240.00
Curriculum and Instruction
6.AF.10
Standard: 6.AF.10 Use variables to represent two quantities in a proportional relationship in a real-world problem; write
an equation to express one quantity, the dependent variable, in terms of the other quantity, the independent variable.
Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to
the equation.
Teacher Background Information:
Verbal Description: In the banana situation, the cost depends on the number of pounds of bananas purchased. In
functional language we can say, “cost is a function of the number of pounds of bananas purchased. The phrase “is a
function of” expresses the dependent relationship. The cost depends on – is a function of the pounds purchased. The
verbal description of the explicit equation for the banana situation might be state as, “You multiply the number of pounds
bought by $2.40.” The verbal description of the explicit equation provides a connection from the context to the
symbolic representation. Students may struggle with using variables, and being able to first describe the equation in
words is an important stepping-stone for being able to use symbols.
Equation:
Suppose that we pick a letter, p, to represent the number of pounds purchased. The cost is represented by the equation c
= (2.4 x p), where c is the letter selected to stand for cost. This equation defines a mathematical relationship between two
values, cost and pounds. By expressing a function as an equation, it is possible to find the cost of any number of pounds
of bananas. Conversely, if Bobby wants to spend, $100, he can figure out how many pounds of bananas he can buy.
It is particularly important that students explain what each number and each variable represent.
Graphs:
Six different values for pounds are plotted on a
graph. The horizontal axis represents the number of
pounds purchased, and the vertical axis, the cost. As
we have already established, the cost goes up as the
pounds purchased goes up. There is, in this
situation, a linear pattern to the six values. In this
context, it means that the cost is going up at a
constant rate of $2.40 per pound. The graphical
representation allows one to see “at a glance” that
the relationship between pounds and cost is linear,
a straight line, and is increasing. It also can be used
to get a quick approximate answer to Bobby’s costs,
such as “How much will it cost Bobby to buy 8
pounds of bananas?” The context gives meaning to
the graph, and the graph adds understanding to the
context.
Indianapolis Public Schools
Curriculum and Instruction
Instructional Week 21
6th Grade Mathematics Assessment
Name:______________________________
1.
y– 4
Solve for y.
5
2
=2
6
3
1
6
1
B) y = 2
2
7
C) y = 6
9
1
D) y = 7
2
A) y = 2
2.
18
=6
d
Solve for d.
A) d =
1
3
B) d = 3
C) d = 12
D) d = 108
Indianapolis Public Schools
Curriculum and Instruction
3.
You have a set of triangles. They all have one side that is 10 centimeters long, which we will consider as
the base of the triangle.
a) The triangles each have a different height (as measured off of the 10-centimeter base) and so
have different areas. Fill in the table below with the missing heights and areas.
Height (centimeters)
Area (square centimeters)
20
25
40
250
b) Plot the ordered pairs from the table in the coordinate plane and label them with their coordinates.
c) Where can you see the answers to part (a) in the coordinate plane?
d) If A represents the area and h represents the corresponding height, write an equation using A
and h that represents the area of any triangle shown by the graph.
______________________________________________________________________
Indianapolis Public Schools
Curriculum and Instruction
Grade 6
Instructional Week 21 Assessment
Answer Key
Question
Correct Answer
1
D
6.AF.5
2
B
6.AF.5
3a
6.AF.10
3b
6.AF.10
3c
The heights from the table are the x-coordinates
of the plotted points. The areas from the table
are the y-coordinates.
3d
2
1
0
Standard(s)
A=
10h
(or other equivalent equation)
2
6.AF.10
6.AF.10
Content Rubric
A score of two indicates a thorough understanding of the mathematical concepts embodied in the
task. The response
 shows content related work executed correctly and completely.
A score of one indicates a partial understanding of the mathematical concepts embodied in the
task. The response
 contains errors in the content related work
A score of zero indicates limited or no understanding of the mathematical concepts embodied in
the task.
Indianapolis Public Schools
Curriculum and Instruction