Core Content Connectors by Common Core State Standards: Mathematics 7th Grade

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Grade 7 Overview

Ratios and Proportional Relationships

• Analyze proportional relationships and use them to solve real-world and mathematical problems.

The Number System

• Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Expressions and Equations

• Use properties of operations to generate equivalent expressions.
• Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Geometry

• Draw, construct and describe geometrical figures and describe the relationships between them.
• Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Statistics and Probability

• Use random sampling to draw inferences about a population.
• Draw informal comparative inferences about two populations.
• Investigate chance processes and develop, use, and evaluate probability models.

Ratios and Proportional Relationships	7.RP
Analyze proportional relationships and use them to solve real-world and mathematical problems.
1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction ½ / ¼ miles per hour, equivalently 2 miles per hour.
CCCs linked to 7.RP.A.1	7.NO.2f3 Find unit rates given a ratio.
	7.PRF.1e1 Determine unit rates associated with ratios of lengths, areas, and other quantities measured in like units.
	7.ME.2e2 Solve one step problems involving unit rates associated with ratios of fractions.
2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
CCCs linked to 7.RP.A.2	7.NO.2f1 Identify the proportional relationship between two quantities.
	7.NO.2f2 Determine if two quantities are in a proportional relationship using a table of equivalent ratios or points graphed on a coordinate plane.
	7.PRF.1e2 Represent proportional relationships on a line graph.
	7.NO.2f4 Use a rate of change or proportional relationship to determine the points on a coordinate plane.
3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
CCCs linked to 7.RP.A.3	7.NO.2h 1 Find percents in real world contexts.
	7.NO.2h2 Solve one step percentage increase and decrease problems
	7.NO.2f5 Use proportions to solve ratio problems
	7.NO.2f6 Solve word problems involving ratios
	7.PRF.1f1 Use proportional relationships to solve multistep percent problems.
	7.NO.2h 1 Find percents in real world contexts.
	7.NO.2h2 Solve one step percentage increase and decrease problems

The Number System	7.NS
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
CCCs linked to 7.NS.A.1	7.NO.1g1 Identify the additive inverse of a number (e.g., -3 and +3).
	7.NO.1g2 Identify the difference between two given numbers on a number line using absolute value.
	8.NO.2i3 Solve one step addition, subtraction, multiplication, division problems with fractions, decimals, and positive/negative numbers.
	8.NO.2i4 Solve two step addition, subtraction, multiplication, and division problems with fractions, decimals, or positive/negative numbers.
2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real world contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
CCCs linked to 7.NS.A.2	7.NO.2i1 Solve multiplication problems with positive/negative numbers.
	7.NO.2i2 Solve division problems with positive/negative numbers.
3. Solve real-world and mathematical problems involving the four operations with rational numbers.
CCCs linked to 7.NS.A.3	8.NO.2i3 Solve one step addition, subtraction, multiplication, division problems with fractions, decimals, and positive/negative numbers
	8.NO.2i4 Solve two step addition, subtraction, multiplication, and division problems with fractions, decimals, or positive/negative numbers

Expressions and Equations	7.EE
Use properties of operations to generate equivalent expressions.
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
CCCs linked to 7.EE.A.1	7.SE.1f3 Add and subtract linear expressions.
	7.SE.1f4 Factor and expand linear expressions.
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."
CCCs linked to 7.EE.A.2	None
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
CCCs linked to 7.EE.B.3	7.PRF.1g1 Solve real-world multi-step problems using whole numbers.
4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
CCCs linked to 7.EE.B.4	7.SE.1f2 Solve equations with 1 variable based on real-world problems.
	7.SE.1f1 Set up equations with 1 variable based on real-world problems.
	7.PRF.1g2 Use variables to represent quantities in a real‐world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
	7.PRF.2d Use a calculator to solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers.

Geometry	7.G
Draw, construct and describe geometrical figures and describe the relationships between them.
1. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
CCCs linked to 7.G.A.1	7.ME.1d1 Solve problems that use proportional reasoning with ratios of length and area.
	7.ME2e1 Solve one step real world problems related to scaling.
2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
CCCs linked to 7.G.A.2	7.GM.1e1 Construct or draw plane figures using properties.
3. Describe the two-dimensional figures that result from slicing three dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
CCCs linked to 7.G.A.3	7.GM.1h5 Describe the two-dimensional figures that result from a decomposed three-dimensional figure.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
CCCs linked to 7.G.B.4	7.ME2d1 Apply formula to measure area and circumference of circles.
5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
CCCs linked to 7.G.B.5	8.GM.1i1 Identify supplementary angles.
	8.GM.1i2 Identify complimentary angles.
	8.GM.1i3 Identify adjacent angles.
	8.GM.1i4 Use angle relationships to find the value of a missing angle.
	6. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
CCCs linked to 7.G.B.6	7.GM.1h1 Add the area of each face of a prism to find surface area of three dimensional objects.
	7.GM.1h2 Find the surface area of three-dimensional figures using nets of rectangles or triangles.
	7.GM.1h3 Find area of plane figures and surface area of solid figures (quadrilaterals).
	7.GM.1h4 Find area of an equilateral, isosceles, and scalene triangle.
	7.ME.2c 1 Solve one step real world measurement problems involving area, volume, or surface area of two- and three-dimensional objects.

Statistics and Probability	7.SP
Use random sampling to draw inferences about a population.
1. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
CCCs linked to 7.SP.A.1	7.DPS.1b1 Determine sample size to answer a given question.
2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
CCCs linked to 7.SP.A.2	7.DPS.1k1 Analyze graphs to determine or select appropriate comparative inferences about two samples or populations.
Draw informal comparative inferences about two populations.
3. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
CCCs linked to 7.SP.B.3	7.DPS.1j1 Make or select a statement to compare the distribution of 2 data sets.
4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
CCCs linked to 7.SP.B.4	7.DPS.1i2 Identify the range (high/low), median(middle), mean, or mode of a given data set.
	7.DPS.1k1 Analyze graphs to determine or select appropriate comparative inferences about two samples or populations.
	7.DPS.1j2 Make or select an appropriate statements based upon two unequal data sets using measure of central tendency and shape.
Investigate chance processes and develop, use, and evaluate probability models.
5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
CCCs linked to 7.SP.C.5	7.DPS.2d1 Describe the probability of events as being certain or impossible, likely, less likely or equally likely.
	7.DPS.2d2 State the theoretical probability of events occurring in terms of ratios (words, percentages, decimals).
6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
CCCs linked to 7.SP.C.6	7.DPS.2d4 Make a prediction regarding the probability of an event occurring; conduct simple probability experiments.
7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny.
CCCs linked to 7.SP.C.7	7.DPS.2d5 Compare actual results of simple experiment with theoretical probabilities.
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
CCCs linked to 7.SP.C.8	7.DPS.2e1 Determine the theoretical probability of multistage probability experiments (2 coins, 2 dice).
	8.DPS.2e4 Determine the theoretical probability of multistage probability experiments (2 coins, 2 dice).
	7.DPS.2e2 Collect data from multistage probability experiments (2 coins, 2 dice)
	8.DPS.2e5 Collect data from multistage probability experiments(2 coins, 2 dice).
	7.DPS.2e3 Compare actual results of multistage experiment with theoretical probabilities.
	8.DPS.2e6 Compare actual results of multistage experiment with theoretical probabilities.