Topic 1: Factors, Multiples, Prime, Composite Numbers
Established Goals:
• Students will apply number theory concepts.
• Students will use factors and multiples to solve real-world problems.
Understandings:
Students will understand that:
• there are relationships among factors, multiples, divisors, and products.
• every counting number is divisible by 1 and itself, and some counting numbers are also
divisible by other numbers.
• there is a relationship between two factors of a product and the dimensions of a rectangle
• there are relationships among factors, multiples, divisors, and products
• the multiplicative structure of numbers, includes the concepts of prime and composite
numbers, evens, odds, and prime factorizations
Essential Questions:
• Will finding the factors help me solve the problem?
• What do the factors and multiples of the numbers tell me about the situation?
• What problems could be solved by finding the common factors and common multiples of
numbers?
Students will know:
• a prime number has only 2 factors, 1 and itself; a composite numbers has more than 2 factors.
• a rectangular array (the area) may be used to represent the product of factor pairs.
• how to find the factors and multiples of a number.
• strategies for finding least common multiples, and greatest common factors.
• every whole number can be written in exactly one way as a product of prime numbers.
Students will be able to:
• apply number theory concepts, including prime factorization, greatest common factor and
least common multiple, to the solution of problems.
• recognize and use properties of prime and composite numbers, even and odd numbers, and
square numbers.
• recognize and use the fact that every whole number can be written in exactly one way as a
product of prime numbers.
• use factors and multiples to solve problems and to explain some numerical facts of everyday
life.
• fluently divide multi-digit numbers using the standard algorithm.
• fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm
for each operation.
• find the greatest common factor of two whole numbers less than or equal to 100 and the least
common multiple of two whole numbers less than or equal to 12.
• 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.
Resource: CMP2 Prime Time
Topic 2: Data Analysis
Established Goals:
• Students will read, understand, interpret, and communicate data as it is critical in modeling a
variety of real-world situations, drawing appropriate inferences, making informed decisions,
and justifying those decisions.
• Students will develop understanding of statistical variability.
Understandings:
Students will understand that:
• The message conveyed by the data depends on how the data is collected, represented, and
summarized.
• The results of a statistical investigation can be used to support or refute an argument.
• a set of data collected to answer a statistical question has a distribution which can be
described by its center, spread, and overall shape.
• a statistical question is one that anticipates variability in the data related to the question and
accounts for it in the answers.
Essential Questions:
How can the collection, organization, interpretation, and display of data be used to answer
questions?
Students will know:
• categorical and numerical data can be collected, organized, described, and displayed in
various ways to answer questions and make comparisons.
• statistical measures will provide useful information about the distribution of data.
• a measure of center for a numerical data set summarizes all of its values with a single
number, while a measure of variation describes how its values vary with a single number.
Students will be able to:
• collect, generate, organize, and display data (categorical and numerical)
• make frequency tables for numerical data, grouping the data in different ways to investigate
how different groupings describe the data, (e.g., relative and cumulative frequency) and use
histograms of the data and the relative frequency distribution to interpret the data
• read, interpret, select, construct, analyze, generate questions about, and draw inferences from
displays of data
1. Bar graph, line graph, circle graph, table, histogram, dot plot, box plot
2. Range, mean, median, and mode
• respond to questions about data, generate their own questions and hypotheses, and formulate
strategies for answering their questions and testing their hypotheses
• give quantitative measures of center (median and/or mean) and variability (interquartile
range and/or mean absolute deviation), as well as describe any overall pattern and any
striking deviations from the overall pattern with reference to the context in which the data
were gathered.
• relate the choice of measures of center and variability to the shape of the data distribution and
the context in which the data were gathered.
Resource: CMP2 Data About Us
Topic 3: Fractions
Established Goals:
• Students will know the relationships between and among fractions, decimals, and percents.
• Students will be able to use multiple representations of parts of a whole.
• Students will extend understanding of fractions and decimals to include place values greater
than hundredths.
Understandings:
Students will understand that:
• the whole must be known in order to compare fractional parts of it.
• the number line can be broken into benchmarks in order to approximate the size of fractions,
decimals, and percents.
• division is an interpretation of fractions.
• decimals as fractions should be connected to decimals as an extension of the place-value
system.
• percents are a part-whole relationship where the whole is not out of 100 but scaled to be “out
of 100.”
Essential Questions:
• When do we need to consider amounts that do not represent whole numbers?
• What models or diagrams might be helpful in understanding the situation and the
relationships among quantities?
• Should a solution be expressed as fractions, decimals, or percents?
• How will knowing equivalent forms of fractions, decimals, or percents help to solve
problems?
• What problems would require a comparison or ordering a set of fractions, decimals, and
percents?
• What does it mean to have more than 100%?
Students will know:
• that benchmarks help to determine relative sizes of fractions.
• how to move flexibly among fraction, decimal, and percent representations.
• why fractions with a denominator of 100 are useful.
• fractions on a number line can represent a location on the number line and also the length
from one point to another on the number line.
Students will be able to:
• order positive and/or negative rational numbers.
• express rational numbers in equivalent forms.
• make estimates and use benchmarks.
• model situations involving fractions, decimals, and percents.
• use equivalent fractions to reason about situations.
• compare and order fractions and decimals.
• use benchmarks that relate different forms of rational numbers (for example, 50% is the same
as 0.5 or 1/2).
• reason with fractions greater than 1 and percents greater than 100%.
Resource: Bits and Pieces I
Topic 4: Geometry
Established Goals:
• Students will find the area of two-dimensional figures.
• Students will solve real-world and mathematical problems involving surface area and
volume.
• Students will draw polygons in a coordinate plane.
• Students will develop spatial sense and understand geometric relationships.
Understandings:
Students will understand that:
• attributes belonging to a category of two dimensional figures also belong to all subcategories
of that category.
• figures can be composed and decomposed into regular shapes.
Essential Questions:
• What applications would make use of a coordinate system as a model?
• How do simple polygons work together to make more complex shapes?
Students will know:
• how to plot points in the coordinate plane.
• the area of triangles and quadrilaterals.
• the volume of a rectangular prism, by formula and by packing it with unit cubes.
• the 2-dimensional pattern for a 3-dimensional shape is called a net.
• a net folds up to be the 3-dimensional shape.
Students will be able to:
• find the area of right triangles, other triangles, special quadrilaterals, and polygons by
composing into rectangles or decomposing into triangles and other shapes.
• apply the techniques of composing/decomposing in the context of solving real-world and
mathematical problems.
• find the volume of a right rectangular prism with fractional edge lengths by packing it with
unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same
as would be found by multiplying the edge lengths of the prism.
• apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with
fractional edge lengths in the context of solving real-world and mathematical problems.
• draw polygons in the coordinate plane given coordinates for the vertices.
• use coordinates to find the length of a side joining points with the same first coordinate or the
same second coordinate. Apply these techniques in the context of solving real-world and
mathematical problems.
• represent three-dimensional figures using nets made up of rectangles and triangles, and use
the nets to find the surface area of these figures. Apply these techniques in the context of
solving real-world and mathematical problems.
Resources:
CMP2 Covering and Surrounding
Supplemental worksheets
Navigating through Geometry (NCTM)
Topic 5: Operations with Fractions
Established Goals:
• Apply and extend previous understandings of multiplication and division to divide fractions
by fractions.
Understandings:
Students will understand that:
• fact families can be used to represent the inverse relationship between addition and
subtraction, and between multiplication and division.
• finding a fraction of a number means multiplication
• the concept of a unit rate a/b is associated with a ratio a:b with b ≠ 0.
Essential Questions:
• What do whole number operations reveal about the meaning of operations with fractions?
• Do results from algorithms support those found with models?
Students will know:
• common denominators are needed to add or subtract fractions.
• the impact of multiplying a number by a number less than or greater than one.
• benchmarks are needed to estimate the sum of fractions and decimals.
• ratio language to describe a ratio relationship between two quantities.
Students will be able to:
• add and subtract fractions with unlike denominators (including mixed numbers) by replacing
given fractions with equivalent fractions in such a way as to produce an equivalent sum or
difference of fractions with like denominators.
• solve word problems involving addition and subtraction of fractions referring to the same
whole, including cases of unlike denominators, e.g., 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.
• interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).
• 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.
• apply and extend previous understandings of multiplication to multiply a fraction or whole
number by a fraction.
• solve real world problems involving multiplication of fractions and mixed numbers, e.g., by
using visual fraction models or equations to represent the problem.
• apply and extend previous understandings of division to divide unit fractions by whole
numbers and whole numbers by unit fractions.
• interpret and compute quotients of fractions, and solve word problems involving division of
fractions by fractions, e.g., by using visual fraction models and equations to represent the
problem.
• use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by
reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or
equations.
•
Resource: Bits and Pieces II
Topic 6: The Number System
Established Goals:
• Students will apply and extend previous understandings of numbers to the system of rational
numbers.
Understandings:
Students will understand that:
• positive and negative numbers are used together to describe quantities having opposite
directions or values (e.g., temperature above/below zero, elevation above/below sea level,
credits/debits, positive/negative electric charge).
• a rational number is a point on the number line.
• rational numbers can be ordered.
• the absolute value of a rational number is its distance from 0 on the number line.
Essential Questions:
• What real-world applications would involve the use of the absolute value of numbers?
• On a number line, how does the location of a positive rational number relate to the location
of the negative rational number with the same absolute value?
Students will know:
• opposite signs of numbers indicates locations on opposite sides of 0 on the number line;
recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3,
and that 0 is its own opposite.
• signs of numbers in ordered pairs indicate locations in quadrants of the coordinate plane.
• when two ordered pairs differ only by signs, the locations of the points are related by
reflections across one or both axes.
Students will be able to:
• use positive and negative numbers to represent quantities in real-world contexts, explaining
the meaning of 0 in each situation.
• extend number line diagrams and coordinate axes familiar from previous grades to represent
points on the line and in the plane with negative number coordinates.
• find and position integers and other rational numbers on a horizontal or vertical number line
diagram.
• find and position pairs of integers and other rational numbers on a coordinate plane.
• interpret statements of inequality as statements about the relative position of two numbers on
a number line diagram.
• write, interpret, and explain statements of order for rational numbers in real-world contexts.
• interpret absolute value as magnitude for a positive or negative quantity in a real-world
situation.
• distinguish comparisons of absolute value from statements about order.
• solve real-world and mathematical problems by graphing points in all four quadrants of the
coordinate plane.
Resource: Supplementary worksheets
Topic 7: Expressions and Equations
Established Goals:
• Students will apply and extend previous understandings of arithmetic to algebraic
expressions.
• Students will reason about and solve one-variable equations and inequalities.
• Students will represent and analyze quantitative relationships between dependent and
independent variables.
Understandings:
Students will understand that:
• variables may represent two quantities in a real-world problem that change in relationship to
one another.
Essential Questions:
• What do whole number operations reveal about the meaning of operations with fractions?
• Do results from algorithms support those found with models?
• Which values from a specified set, if any, make the equation or inequality true?
Students will know:
• in expressions, variables represent numbers
• a variable can represent an unknown number or any number in a specified set.
• equations may be written to express one quantity, thought of as the dependent variable, in
terms of the other quantity, thought of as the independent variable.
Students will be able to:
• write and evaluate numerical expressions involving whole-number exponents.
• write expressions that record operations with numbers and with letters standing for numbers.
• 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.
• 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).
• apply the properties of operations to generate equivalent expressions.
• identify when two expressions are equivalent (i.e., when the two expressions name the same
number regardless of which value is substituted into them).
• use substitution to determine whether a given number in a specified set makes an equation or
inequality true.
• solve real-world and mathematical problems by writing and solving equations of the form
x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
• write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have
infinitely many solutions; represent solutions of such inequalities on number line diagrams.
• analyze the relationship between the dependent and independent variables using graphs and
tables, and relate these to the equation.
Resource:
Supplementary worksheets
Apple Pi – ratio of circumference to diameter
http://illuminations.nctm.org/LessonDetail.aspx?ID=L573
Fraction/decimal/percent
http://illuminations.nctm.org/ActivityDetail.aspx?ID=45
The Factor Game http://illuminations.nctm.org/ActivityDetail.aspx?ID=12 Histogram maker http://illuminations.nctm.org/ActivityDetail.aspx?ID=78 The Product Game http://illuminations.nctm.org/ActivityDetail.aspx?ID=29 Area of rectangle, parallelogram, triangle http://illuminations.nctm.org/ActivityDetail.aspx?ID=21
Box-whiskers
http://illuminations.nctm.org/LessonDetail.aspx?ID=L231