Math Worksheet for 15-Year-Olds: Solving Linear Equations, Data Analysis, and Linear Graphs

Instructions

Complete all sections of the worksheet to the best of your ability. Show your working in the space provided. You may use a calculator for sections 2 and 3. For graphing questions, use the grids provided.

Section 1: Solving Linear Equations

Solve the following equations for the unknown variable. Show your steps.

1. 5x - 3 = 22

   
   
   

2. a/4 + 7 = 10

   
   
   

3. 2(y - 3) = 14

   
   
   

4. (2x + 5) / 3 = 7

   
   
   

5. (m - 10) / 6 = -2

   
   
   

6. 15 - 2b = 3

   
   
   

Section 2: Data Analysis

Part A: Summary Statistics

A teacher recorded the results for a recent quiz for two different classes. The results (out of 20) are shown below.

Class A: 12, 15, 16, 16, 17, 18, 20

Class B: 10, 14, 15, 16, 16, 19, 20

1. For Class A, calculate the:

   • Mean: ____________
   • Median: ____________
   • Mode: ____________
   • Range: ____________

2. For Class B, calculate the:

   • Mean: ____________
   • Median: ____________
   • Mode: ____________
   • Range: ____________

3. Write one or two sentences comparing the performance of the two classes, using your calculated statistics to support your conclusion.

   
   
   

Part B: Two-Variable Data

A student surveyed their friends to see if there was a relationship between the hours they spent playing video games and the hours they spent on homework each week. The results are in the table below.

1. Plot the data on the scatter plot grid below. The horizontal axis (x-axis) should be 'Hours Gaming' and the vertical axis (y-axis) should be 'Hours Homework'.

   Scatter Plot

   16
   14
   12
   10
   8
   6
   4
   2
   0
   	0	2		4		6		8		10		12		14		16

2. Describe the relationship (correlation) between the two variables. Is it positive, negative, or is there no correlation? Explain your answer.

   
   
   

Section 3: Linear Relationships

Part A: Midpoint, Gradient, and Length

You are given two points: P(-2, 3) and Q(4, -5).

Use the following formulas to find the properties of the line segment PQ:

• Gradient (m): (y₂ - y₁) / (x₂ - x₁)
• Midpoint: ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
• Length/Distance: √((x₂ - x₁)² + (y₂ - y₁)²)

1. Calculate the gradient of the line segment PQ.

   
   
   

2. Find the coordinates of the midpoint of PQ.

   
   
   

3. Calculate the length of PQ. You can leave your answer as a simplified surd (e.g., 2√5) or round to two decimal places.

   
   
   

Part B: Graphing a Linear Equation

Consider the linear equation: y = -2x + 3

1. Complete the table of values below for the equation.

   x	-2	-1	0	1	2
   y

2. Plot the points from your table onto the Cartesian plane below and draw a straight line through them.

   Cartesian Plane

   5
   0
   -5
   	-4		-2		0		2		4

Answer Key

Section 1: Solving Linear Equations

1. x = 5 (5x = 25, x = 25/5)
2. a = 12 (a/4 = 3, a = 3 * 4)
3. y = 10 (y - 3 = 7, y = 7 + 3)
4. x = 8 (2x + 5 = 21, 2x = 16, x = 16/2)
5. m = -2 (m - 10 = -12, m = -12 + 10)
6. b = 6 (-2b = -12, b = -12/-2)

Section 2: Data Analysis

Part A:

1. Class A:
   • Mean: (12+15+16+16+17+18+20) / 7 = 114 / 7 ≈ 16.29
   • Median: 16 (the middle value)
   • Mode: 16 (the most frequent value)
   • Range: 20 - 12 = 8
2. Class B:
   • Mean: (10+14+15+16+16+19+20) / 7 = 110 / 7 ≈ 15.71
   • Median: 16 (the middle value)
   • Mode: 16 (the most frequent value)
   • Range: 20 - 10 = 10
3. Comparison: (Sample Answer) On average, Class A performed slightly better than Class B, as shown by its higher mean score (16.29 vs 15.71). Both classes have the same median and mode (16), suggesting the typical student in both classes achieved a similar result. However, Class B's scores were more spread out, as indicated by its larger range.

Part B:

1. Scatter Plot: The student should plot the following coordinates: (10, 4), (8, 6), (5, 8), (15, 2), (2, 10), (12, 3), (0, 12).
2. Correlation: (Sample Answer) There is a strong negative correlation between the hours spent gaming and the hours spent on homework. As the number of hours spent gaming increases, the number of hours spent on homework tends to decrease. The points on the plot form a rough line that goes down from left to right.

Section 3: Linear Relationships

Part A: (Points P(-2, 3) and Q(4, -5))

1. Gradient: m = (-5 - 3) / (4 - (-2)) = -8 / 6 = -4/3
2. Midpoint: ( (-2 + 4)/2 , (3 + (-5))/2 ) = ( 2/2 , -2/2 ) = (1, -1)
3. Length: √((4 - (-2))² + (-5 - 3)²) = √(6² + (-8)²) = √(36 + 64) = √100 = 10 units

Part B: (Equation y = -2x + 3)

1. Table of values:

   x	-2	-1	0	1	2
   y	7	5	3	1	-1

2. Graph: The student should plot the points (-2, 7), (-1, 5), (0, 3), (1, 1), and (2, -1). The points should form a straight line that passes through the y-axis at 3 and has a negative gradient.