Focus Areas for Seventh Grade
- Developing understanding of proportional relationships
- Operating with rational numbers
- Working with expressions and linear equations
- Solving problems involving scale drawings and geometric constructions
- Drawing inferences about populations based on samples
Math Standards by Domain
Ratios and Proportional Relationships (7.RP)
What Students Learn:
Students develop a deep understanding of proportional relationships, learning to compute unit rates associated with ratios of fractions, recognize proportional relationships from tables, graphs, equations, and verbal descriptions, and use proportional reasoning to solve multi-step ratio and percent problems. This includes understanding constant of proportionality and representing it in different forms.
Key Skills:
- Compute unit rates including those involving complex fractions (e.g., 1/2 mile in 1/4 hour = 2 mph)
- Recognize proportional relationships in tables (constant ratio between quantities)
- Identify proportional relationships in graphs (straight line through origin)
- Determine the constant of proportionality (unit rate) from tables, graphs, equations, or descriptions
- Represent proportional relationships using equations in the form y = kx
- Explain what a point (x, y) on a graph of a proportional relationship means
- Solve multi-step ratio and percent problems including simple interest, tax, markups, markdowns, tips, and commissions
If 3/4 pound of rice costs $2, what is the unit rate (price per pound)? $2 ÷ 3/4 = $2 × 4/3 = $8/3 or about $2.67 per pound
Example (Proportional Relationship):Distance traveled at constant speed: y = 55x where y = distance in miles, x = time in hours, and 55 mph is the constant of proportionality
Example (Percent Problem):A jacket originally costs $80. It's on sale for 25% off. What is the sale price? Discount: 0.25 × $80 = $20. Sale price: $80 - $20 = $60
The Number System (7.NS)
What Students Learn:
Students extend their understanding of arithmetic to include all rational numbers (positive and negative fractions, decimals, and integers). They learn to add, subtract, multiply, and divide rational numbers in any form, understand properties of operations, and apply these skills to solve complex real-world and mathematical problems. This includes working with negative numbers conceptually and computationally.
Key Skills:
- Apply properties of operations as strategies to add and subtract rational numbers
- Understand that subtracting is the same as adding the additive inverse: p - q = p + (-q)
- Add and subtract integers using number lines and absolute value
- Add and subtract fractions and decimals with unlike signs
- Understand that distance between two numbers on a number line is the absolute value of their difference
- Apply properties of operations to multiply and divide rational numbers
- Understand that multiplying/dividing by -1 changes the sign: (-1)(-1) = 1
- Convert between fractions, decimals, and percents to facilitate operations
- Solve multi-step real-world problems involving all four operations with rational numbers
-5 + 8 = 3 (start at -5, move 8 units right)
3/4 + (-1/2) = 3/4 - 1/2 = 3/4 - 2/4 = 1/4
Example (Multiplication):-3 × 4 = -12 (negative times positive equals negative)
-3 × -4 = 12 (negative times negative equals positive)
Example (Division):-15 ÷ 3 = -5; -15 ÷ -3 = 5
Example (Real-World):A submarine is at -150 feet (150 feet below sea level). It descends another 75 feet. What is its new position? -150 + (-75) = -225 feet
Expressions and Equations (7.EE)
What Students Learn:
Students learn to manipulate algebraic expressions using properties of operations including distributive, associative, and commutative properties. They understand how to add, subtract, factor, and expand linear expressions with rational coefficients, and recognize when expressions are equivalent. Students also learn to rewrite expressions in different forms to solve problems more efficiently.
Key Skills:
- Apply properties to add and subtract linear expressions (combine like terms)
- Factor expressions using the distributive property: 6x + 15 = 3(2x + 5)
- Expand expressions: 4(3x - 2) = 12x - 8
- Simplify expressions with rational coefficients: 1/2(4x + 6) - 3x = 2x + 3 - 3x = -x + 3
- Understand that rewriting an expression can reveal different properties
- Recognize equivalent expressions
- Use properties strategically to simplify or rewrite expressions
3x + 5 - 2x + 7 = (3x - 2x) + (5 + 7) = x + 12
Example (Distributive Property):Expand: -2(3x - 4) = -6x + 8
Factor: 8x + 12 = 4(2x + 3)
Example (Rational Coefficients):2/3(6x - 9) + 4x = 4x - 6 + 4x = 8x - 6
What Students Learn:
Students learn to solve multi-step real-world and mathematical problems involving rational numbers in any form (whole numbers, fractions, decimals). They apply properties of operations, use variables to represent unknown quantities, write and solve equations and inequalities, and construct viable arguments about solutions. This includes solving equations of the form px + q = r and p(x + q) = r.
Key Skills:
- Solve multi-step problems involving addition, subtraction, multiplication, and division of rational numbers
- Convert between forms (fractions, decimals, percents) to solve problems
- Assess the reasonableness of answers using estimation and mental computation
- Use variables to represent quantities in real-world or mathematical problems
- Construct simple equations and inequalities to solve problems
- Solve word problems leading to equations of the form px + q = r and p(x + q) = r
- Solve word problems leading to inequalities of the form px + q > r or px + q < r
- Graph the solution set of inequalities and interpret in context
A cell phone plan costs $25/month plus $0.10 per text message. Last month the bill was $37. How many texts were sent? 25 + 0.10t = 37, so 0.10t = 12, therefore t = 120 texts
Example (Equation):3(x + 2) = 15. Solve: 3x + 6 = 15, so 3x = 9, therefore x = 3
Example (Inequality):You have $50 to spend on party supplies. Decorations cost $12. Pizzas cost $8 each. How many pizzas can you buy? 12 + 8p ≤ 50, so 8p ≤ 38, therefore p ≤ 4.75. You can buy at most 4 pizzas.
Geometry (7.G)
What Students Learn:
Students learn to work with scale drawings, understanding how to compute actual lengths and areas from a scale drawing and create scale drawings using given scale factors. They use geometric tools to draw geometric shapes with given conditions, focusing on triangles with specific angle or side measurements. Students also learn to describe and analyze two-dimensional figures that result from slicing three-dimensional figures.
Key Skills:
- Solve problems involving scale drawings of geometric figures
- Compute actual lengths and areas from a scale drawing using the scale factor
- Reproduce a scale drawing at a different scale
- Draw geometric shapes with given conditions using rulers, protractors, and compasses
- Focus on constructing triangles from three measures of angles or sides
- Understand when given conditions determine a unique triangle, more than one triangle, or no triangle
- Describe two-dimensional figures that result from slicing three-dimensional figures (cross-sections)
- Visualize and identify cross-sections of pyramids, prisms, cylinders, cones, and spheres
A map has a scale of 1 inch : 50 miles. Two cities are 3.5 inches apart on the map. What is the actual distance? 3.5 × 50 = 175 miles
Example (Triangle Construction):Draw a triangle with sides 3 cm, 4 cm, and 5 cm using a ruler and compass. This creates one unique triangle.
Can you draw a triangle with sides 2 cm, 3 cm, and 10 cm? No, because 2 + 3 < 10 (triangle inequality)
Example (Cross-Sections):If you slice a rectangular pyramid horizontally, the cross-section is a rectangle. If you slice it vertically through the apex, you get a triangle.
What Students Learn:
Students learn about the relationships in circles, including the formulas for circumference (C = πd or C = 2πr) and area (A = πr²). They solve problems involving supplementary, complementary, vertical, and adjacent angles using equations. Students also learn to solve real-world problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Key Skills:
- Understand the relationships among radius, diameter, and circumference of a circle
- Use formulas for circumference: C = πd or C = 2πr
- Use formulas for area of a circle: A = πr²
- Apply circle formulas to solve real-world problems
- Understand and identify supplementary angles (sum to 180°), complementary angles (sum to 90°), vertical angles (equal), and adjacent angles
- Write and solve equations to find unknown angles in geometric figures
- Solve real-world problems involving area and perimeter of two-dimensional figures
- Solve real-world problems involving volume and surface area of three-dimensional objects (cubes, right prisms)
- Work with composite figures made from triangles, quadrilaterals, and other polygons
A circle has a radius of 7 cm. Find its circumference and area.
C = 2πr = 2π(7) = 14π ≈ 43.98 cm
A = πr² = π(7)² = 49π ≈ 153.94 cm²
Example (Angles):Two angles are supplementary. One angle is 3x and the other is x + 20. Find both angles.
3x + (x + 20) = 180, so 4x + 20 = 180, thus 4x = 160, x = 40
Angles are 3(40) = 120° and 40 + 20 = 60°
Example (Volume):A rectangular prism has length 8 cm, width 5 cm, and height 3 cm. Find its volume.
V = l × w × h = 8 × 5 × 3 = 120 cm³
Statistics and Probability (7.SP)
What Students Learn:
Students learn the fundamental concepts of statistical sampling and how to make inferences about a population based on a sample. They understand that generalizations about a population from a sample are valid only if the sample is representative of that population, and learn to use random sampling to produce representative samples. Students also learn to draw informal comparative inferences about two populations.
Key Skills:
- Understand that statistics can be used to gain information about a population by examining a sample
- Recognize that random sampling tends to produce representative samples and valid inferences
- Understand what makes a sample biased or unbiased
- Use data from a random sample to draw inferences about a population
- Estimate population characteristics from sample data
- Generate multiple samples from the same population to gauge variation in estimates
- Understand margin of error and sampling variability informally
To find out what students think about a new lunch menu, which is better: surveying only 7th graders, or surveying random students from all grades? Random students from all grades provides a more representative sample of the entire student population.
Example (Making Inferences):In a random sample of 50 students at a school of 500 students, 30 students prefer online learning. Estimate how many students in the whole school prefer online learning: (30/50) × 500 = 300 students
Example (Biased Sample):Surveying people outside a gym about exercise habits would be biased because gym-goers are more likely to exercise regularly than the general population.
What Students Learn:
Students learn to informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between centers by expressing it as a multiple of a measure of variability. They use measures of center (mean, median) and measures of variability (range, interquartile range, mean absolute deviation) to draw comparative inferences about two populations based on random samples from those populations.
Key Skills:
- Calculate and interpret measures of center: mean (average) and median (middle value)
- Calculate and interpret measures of variability: range, interquartile range (IQR), mean absolute deviation (MAD)
- Understand how outliers affect measures of center and variability
- Compare two data distributions visually using dot plots, histograms, and box plots
- Assess the degree of overlap between two distributions
- Express the difference between centers as a multiple of the measure of variability
- Make informal comparative inferences about two populations based on sample data
- Understand when differences between populations are meaningful versus due to sampling variability
Test scores: 75, 80, 85, 85, 90, 95, 100
Mean = (75+80+85+85+90+95+100)/7 = 87.14
Median = 85 (middle value when ordered)
Example (Measures of Variability):Range = 100 - 75 = 25
IQR = Q3 - Q1 = 95 - 80 = 15
Example (Comparing Populations):Class A mean test score: 82, MAD: 5. Class B mean test score: 78, MAD: 5.
The difference in means is 4 points, which is less than 1 MAD (5 points), suggesting the difference might not be very significant.
What Students Learn:
Students develop a formal understanding of probability, learning that the probability of an event is a number between 0 and 1 that expresses the likelihood of the event occurring. They collect data from probability experiments, compare theoretical and experimental probabilities, and develop probability models. Students also learn to find probabilities of compound events using organized lists, tables, tree diagrams, and simulation, understanding concepts like sample space and independence.
Key Skills:
- Understand that probability is a number from 0 to 1 (0 = impossible, 1 = certain)
- Approximate the probability of a chance event by collecting data
- Develop a probability model by observing frequencies in data generated from a chance process
- Compare theoretical probabilities (what should happen) with experimental probabilities (what actually happened)
- Develop uniform probability models: P(event) = (number of favorable outcomes)/(total number of outcomes)
- Find probabilities of compound events using organized lists, tables, and tree diagrams
- Understand the sample space (all possible outcomes) for compound events
- Design and use simulations to generate frequencies for compound events
- Understand independent events and calculate their probabilities
What is the probability of rolling a 4 on a standard die?
P(4) = 1/6 (1 favorable outcome out of 6 possible outcomes)
Example (Compound Events):What is the probability of flipping two coins and getting two heads?
Sample space: {HH, HT, TH, TT} - 4 equally likely outcomes
P(two heads) = 1/4
Example (Tree Diagram):A bag contains 2 red marbles and 3 blue marbles. You draw one marble, replace it, then draw another. What's the probability of drawing red then blue?
P(red) = 2/5, P(blue) = 3/5
P(red then blue) = 2/5 × 3/5 = 6/25
Example (Experimental vs Theoretical):Theoretical: P(heads) = 1/2. After flipping 100 times, you get 47 heads. Experimental probability = 47/100 = 0.47, which is close to the theoretical 0.5