Subjects › Grade 7 Math
Grade 7 Math: proportions, negative numbers, equations, and geometry
This course follows the Grade 7 Common Core math standards: proportional relationships and unit rates, operations with positive and negative rational numbers, linear expressions and equations, scale drawings and geometric constructions, angle relationships, area and volume including circles, and statistics and probability including sampling, comparing distributions, and compound events. It builds directly on Grade 6 work with fractions, decimals, ratios, and basic geometry, and stops before functions and the Pythagorean theorem, which belong to Grade 8.
A lesson here starts from something the student can already do and moves one question at a time. If a student says a negative times a negative is negative, the tutor doesn't just correct it - it asks what happens to a pattern like 3 x (-2), 2 x (-2), 1 x (-2) as the first factor keeps dropping by 1, and lets the student notice the rule. The tutor checks both whether a student can explain an idea, like why a percent increase is the same as multiplying by 1.05, and whether they can do it, like solving 0.05a + a = goal for a.
Who it is for
- Students taking Grade 7 Common Core math in school, for extra practice or to get unstuck on homework
- Students who are solid on Grade 6 fractions, decimals, and ratios but new to negative number operations
- Parents who want a check on whether their child can explain a method, not just get an answer
- Homeschooling families following the Common Core Grade 7 sequence
- Students preparing for Grade 8, who need these topics solid before functions and the Pythagorean theorem
What you will be able to do
- Solve multistep ratio and percent problems, including discounts, tax, and tips
- Add, subtract, multiply, and divide positive and negative rational numbers, including fractions and decimals
- Write and solve equations and inequalities of the form px + q = r and px + q > r, and graph the solutions
- Compute circumference, area, surface area, and volume of circles, polygons, prisms, and pyramids
- Use a random sample to make an inference about a population and judge whether a sample is biased
- Find probabilities of compound events using tree diagrams, organized lists, and simulation
A taste of a lesson on Multiplying rational numbers and the sign rules
The tutor asks, you answer, and it works from what you actually said. No videos to sit through.
The 32 topics in Grade 7 Math
Each topic builds on the ones before it. The tutor checks what you already know and starts there.
Unit rates with fractions
Compute unit rates from ratios of fractions and of quantities with like or different units (e.g., 1/2 mile in each 1/4 hour gives 2 miles per hour). Students learn to divide the numerator and denominator by the same quantity and to convert units within a rate. Real understanding means explaining what the unit rate tells you about the situation.
Recognizing proportional relationships
A proportional relationship has a constant ratio between quantities, equivalently y = kx with k the constant of proportionality (unit rate). Teach how to test for proportionality in tables (equivalent ratios), graphs (straight line through the origin), and equations, and to identify the constant of proportionality in each representation. Students should be able to explain why a line not through the origin is not proportional.
Builds on Unit rates with fractions
Graphs of proportional relationships and their points
Represent proportional relationships with equations and graphs on a coordinate plane. Interpret what points (x, y) mean in context, especially (0, 0) (no quantity gives no total) and (1, r) (one unit gives the unit rate r). Students should connect the slope of the line to the constant of proportionality.
Builds on Recognizing proportional relationships
Multistep ratio and percent problems
Use proportional reasoning to solve multistep problems involving ratios, percent increase and decrease, discounts, tax, tips, commission, and percent of a quantity. Key idea: convert percents to decimal multipliers (e.g., 5% increase is multiply by 1.05). Students should check that answers are reasonable and explain each step in context.
Builds on Graphs of proportional relationships and their points
Adding rational numbers on the number line
Extend addition to positive and negative rational numbers. Represent p + q as the point a distance |q| from p, moving right for positive q and left for negative q. Explain opposite quantities combining to zero (for example, a hydrogen atom has zero charge), and show that a number and its opposite sum to 0 (additive inverses). Use sign rules for adding numbers with same and different signs.
Builds on Unit rates with fractions
Subtracting rational numbers and distance
Understand subtraction as adding the additive inverse: p - q = p + (-q). Apply this to compute differences of positive and negative rationals, including decimals and fractions. Show that the distance between two rational numbers on the number line equals |p - q| and use this in real contexts such as temperature change or elevation difference.
Builds on Adding rational numbers on the number line
Properties of operations for adding and subtracting rationals
Use commutative, associative, and identity properties as strategies to add and subtract rational numbers efficiently, such as grouping numbers that sum to whole numbers or regrouping fractions with common denominators. Students should justify each rearrangement and check results with estimation.
Builds on Subtracting rational numbers and distance
Multiplying rational numbers and the sign rules
Extend multiplication to rational numbers so that properties (especially the distributive property) still hold, which explains why (-1)(-1) = 1 and why a negative times a negative is positive. Compute products of fractions, decimals, and mixed signs fluently and interpret the sign of the product.
Builds on Subtracting rational numbers and distance
Dividing rational numbers and decimal conversion
Explain that division is the inverse of multiplication and that every quotient of integers (nonzero divisor) is rational; show -(p/q) = (-p)/q = p/(-q) and apply the sign rules to division. Also convert rationals to decimals using long division, and recognize that decimals terminate or eventually repeat.
Builds on Multiplying rational numbers and the sign rules
Properties as strategies for multiplying and dividing rationals
Use the properties of operations (commutative, associative, distributive, reciprocals) to multiply and divide rational numbers efficiently and check answers. Students should rewrite division as multiplication by the reciprocal and reorder factors to simplify computation.
Builds on Dividing rational numbers and decimal conversion
Multi-step word problems with rational numbers
Solve real-world and mathematical multi-step problems using all four operations on positive and negative rational numbers in fraction, decimal, and whole-number forms, choosing tools (mental math, paper, calculator) strategically. Students should interpret results in context and check reasonableness.
Builds on Properties of operations for adding and subtracting rationals, Properties as strategies for multiplying and dividing rationals
Equivalent linear expressions
Apply properties of operations to add, subtract, factor, and expand linear expressions with rational coefficients (e.g., 3(x + 1/2) = 3x + 3/2; 6x + 4 = 2(3x + 2)). Students should combine like terms and explain why two forms are equivalent.
Builds on Properties of operations for adding and subtracting rationals, Properties as strategies for multiplying and dividing rationals
Rewriting expressions to reveal meaning in context
Rewrite expressions in different forms to shed light on a problem: for example, a + 0.05a = 1.05a shows that a 5% increase is multiplying by 1.05; a total cost written as 2(n + 3) reveals a structure. Students should choose the form that answers the question asked.
Builds on Multistep ratio and percent problems, Equivalent linear expressions
Using variables to model quantities
Assign variables to unknown quantities in real-world and math problems, define them clearly with units, and write expressions, equations, and inequalities that represent the relationships among them. Students should reason about what the quantities mean before writing symbols.
Builds on Rewriting expressions to reveal meaning in context
Solving equations px + q = r and p(x + q) = r
Solve word problems leading to equations of the form px + q = r and p(x + q) = r with rational p, q, r, fluently by inverse operations or by undoing steps. Compare the algebraic solution to an arithmetic (work-backward) solution and explain which method is more efficient for a given problem.
Builds on Multi-step word problems with rational numbers, Using variables to model quantities
Solving and graphing inequalities px + q > r and px + q < r
Solve word problems leading to inequalities of the form px + q > r or px + q < r with rational p, q, r. Graph the solution set on a number line using open or closed circles, and interpret the solution in context (for example, the set of possible weekly hours that keep a budget under a limit). Note that dividing by a negative reverses the inequality.
Builds on Solving equations px + q = r and p(x + q) = r
Scale drawings
Solve problems involving scale drawings of geometric figures: compute actual lengths and areas from a drawing with a given scale (such as 1 cm : 5 m), and reproduce a drawing at a different scale. Students should set up a proportion from the scale and check units when converting areas (squared scale factor).
Builds on Graphs of proportional relationships and their points, Multistep ratio and percent problems
Drawing triangles from given conditions
Draw geometric shapes with given conditions using freehand, ruler and protractor, and technology, focusing on triangles built from three side lengths or three angle measures. Students should notice when conditions determine a unique triangle, when they determine more than one, and when no triangle is possible (for example, the triangle inequality for sides, and angles must sum to 180 degrees).
Builds on Scale drawings
Cross-sections of three-dimensional figures
Describe the two-dimensional figures that result from slicing right rectangular prisms and right rectangular pyramids with a plane, including slices parallel to a base (same shape as base) and slices through faces, edges, or diagonally (rectangles, triangles, trapezoids, and other shapes). Students should visualize and sketch these cross-sections.
Builds on Drawing triangles from given conditions
Angle relationships and equations
Use facts about supplementary angles (sum to 180 degrees), complementary angles (sum to 90 degrees), vertical angles (equal), and adjacent angles (add to a larger angle) in multi-step figures to write and solve simple equations for an unknown angle.
Builds on Using variables to model quantities, Solving equations px + q = r and p(x + q) = r
Circumference and area of circles
Know and use C = 2 pi r (or pi d) and A = pi r^2 to solve problems. Give an informal derivation of the relationship between circumference and area, for example by slicing a circle into sectors and rearranging them into an approximate parallelogram whose base is half the circumference and height is the radius. Use pi approximations like 3.14 or 22/7 appropriately.
Builds on Unit rates with fractions, Properties as strategies for multiplying and dividing rationals
Area of triangles, quadrilaterals, and polygons
Solve real-world and mathematical problems involving area of figures composed of triangles, quadrilaterals, and polygons by decomposing them into simpler shapes and using base-height formulas (triangle one-half bh, trapezoid, parallelogram). Students should justify decompositions and check units.
Builds on Properties as strategies for multiplying and dividing rationals, Equivalent linear expressions
Volume and surface area of prisms, pyramids, and composite solids
Solve problems involving volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Use V = Bh for right prisms and the net of a solid to find surface area. Students should decompose composite solids and track units cubed and squared.
Builds on Circumference and area of circles, Area of triangles, quadrilaterals, and polygons
Samples and populations
Understand that statistics can give information about a population by examining a sample, and that generalizations are valid only when the sample is representative. Identify biased samples (e.g., surveying only one lunch table) and explain how random selection helps. Relate sample size to reliability.
Builds on Unit rates with fractions
Inference from random samples and sampling variation
Use data from a random sample to draw inferences about a population with an unknown characteristic (for example, estimating the proportion of red marbles in a bag). Generate multiple samples or simulated samples of the same size to see how estimates vary, and use that variation to judge how reliable a prediction is.
Builds on Multistep ratio and percent problems, Samples and populations
Comparing two data distributions
Informally assess the visual overlap of two numerical data distributions with similar variability by measuring the difference between their centers as a multiple of a measure of variability (e.g., the mean height difference is 3 times the MAD). Use measures of center (mean, median) and variability (range, MAD, IQR) from random samples to draw informal comparative inferences, such as deciding whether two word-length samples differ meaningfully.
Builds on Inference from random samples and sampling variation
Chance events and the probability scale
Understand that the probability of a chance event is a number from 0 to 1 expressing its likelihood; 0 is impossible, 1 is certain, and 1/2 is equally likely to occur or not. Place events on a probability line and classify them as unlikely, likely, or neither. Also know that probabilities of all outcomes in a model sum to 1.
Builds on Unit rates with fractions
Experimental probability and long-run relative frequency
Approximate the probability of a chance event by collecting data on the chance process and observing its long-run relative frequency, such as tossing a coin many times. Predict approximate counts from a given probability (e.g., 0.3 of 200 trials is about 60). Recognize that small samples can vary widely from the model.
Builds on Inference from random samples and sampling variation, Chance events and the probability scale
Uniform probability models
Develop a uniform probability model by assigning equal probability to all outcomes (fair coins, fair dice, random selection from a class) and use it to find probabilities of events as the number of favorable outcomes divided by the total number of equally likely outcomes.
Builds on Multistep ratio and percent problems, Chance events and the probability scale
Non-uniform probability models from data
Develop a probability model that is not uniform by observing frequencies in data from a chance process (e.g., a spinning penny landing heads or a thumbtack landing point up). Compare the model's probabilities with observed frequencies and explain possible sources of discrepancy when agreement is poor.
Builds on Experimental probability and long-run relative frequency, Uniform probability models
Compound events: sample spaces and probabilities
Find probabilities of compound events (such as rolling two dice, two coin flips, or drawing two items) using organized lists, tables, and tree diagrams. Understand that the probability of a compound event is the fraction of outcomes in the sample space for which the event occurs; identify outcomes described in everyday language (e.g., rolling double sixes).
Builds on Uniform probability models
Simulation of compound events
Design and use a simulation (random digits, coins, spinners, or random number generators) to generate frequencies for compound events and approximate their probabilities, such as: if 40% of donors have type A blood, estimate the probability that the next 3 donors include at least one type A. Compare simulated results to the exact model.
Builds on Non-uniform probability models from data, Compound events: sample spaces and probabilities
Practice questions in Grade 7 Math
A few of the kinds of questions you will work through, with answers.
A runner covers 3/4 mile in 1/3 hour. What is the unit rate in miles per hour?
See the answer
2.25 miles per hour
Divide 3/4 by 1/3: (3/4) x (3/1) = 9/4 = 2.25 miles per hour.
What is (-8) x (-0.5)?
See the answer
4
A negative times a negative is positive, and 8 x 0.5 = 4.
Solve for x: 3(x - 2) = 15
See the answer
x = 7
Divide both sides by 3 to get x - 2 = 5, then add 2 to get x = 7.
A circle has radius 4 cm. What is its area in terms of pi?
See the answer
16 pi cm²
Area equals pi r², and 4² = 16, so the area is 16 pi square centimeters.
A coin is flipped and a six-sided die is rolled. What is the probability of getting heads and a number greater than 4?
See the answer
1/6
There are 12 equally likely outcomes; heads with a 5 or 6 gives 2 favorable outcomes, so the probability is 2/12 = 1/6.
Two classes take the same quiz. Class A has a mean of 78 with mean absolute deviation 4, Class B has a mean of 86 with the same spread. How many MADs apart are the means?
See the answer
2 MADs
The difference in means is 8, and 8 divided by the MAD of 4 is 2.
Questions about this subject
Does this course cover the Pythagorean theorem or functions?
No. This course stops before those topics, which are part of Grade 8. It ends with probability and simulation of compound events.
My child struggles with negative numbers specifically. Can they just work on that?
Yes. A student can focus on the number system topics, like adding and subtracting rational numbers on a number line, without working through the whole sequence in order, though some topics do build on earlier ones.
How does the tutor handle a student who gives a wrong answer?
It asks a follow-up question based on what the student said, often a simpler or related case, to help them find the error themselves rather than just stating the correct answer.
Is this aligned to the Common Core standards?
Yes, the topics and their order follow the Grade 7 Common Core math standards for ratios and proportions, the number system, expressions and equations, geometry, and statistics and probability.
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