geometry problem solving grade 8

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Printable 8th Grade Geometry Worksheets

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Common Core - State Standards Initiative

Students learn the basic concepts of algebra, geometry, and graphing before entering high school and learning more complicated topics. It is crucial that students completely grasp these concepts before going on to harder topics like trigonometry or calculus. With our on-demand videos students have the opportunity to go over math problems with a math teacher who knows how to break it down in an easily digestible format.

• Teachers go over definitions along with multiple problems for the skills so that students fully grasp the concepts.
• Skills available for statistics, time, ratios, and other eighth grade skills.
• Students learn how to use the scracthpad to better understand how to tackle the problems.

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Eighth grade math

IXL offers hundreds of eighth grade math skills to explore and learn! Not sure where to start? Go to your personalized Recommendations wall to find a skill that looks interesting, or select a skill plan that aligns to your textbook, state standards, or standardized test.

A. Integers

• 1 Compare and order integers
• 2 Integer addition and subtraction rules
• 3 Add and subtract integers using counters
• 4 Add and subtract integers
• 5 Add and subtract three or more integers
• 6 Add and subtract integers: word problems
• 7 Integer multiplication and division rules
• 8 Multiply and divide integers
• 9 Evaluate numerical expressions involving integers

B. Rational numbers

• 1 Convert between repeating decimals and fractions
• 2 Convert between decimals and fractions or mixed numbers
• 3 Compare rational numbers
• 4 Put rational numbers in order
• 5 Reciprocals and multiplicative inverses
• 6 Add and subtract rational numbers
• 7 Add and subtract rational numbers: word problems
• 8 Apply addition and subtraction rules
• 9 Multiply and divide rational numbers
• 10 Multiply and divide rational numbers: word problems
• 11 Apply multiplication and division rules
• 12 Apply addition, subtraction, multiplication, and division rules
• 13 Evaluate numerical expressions involving rational numbers
• 14 Multi-step word problems

C. Exponents

• 1 Understanding exponents
• 2 Evaluate powers
• 3 Solve equations with variable exponents
• 4 Powers with negative bases
• 5 Powers with decimal and fractional bases
• 6 Understanding negative exponents
• 7 Evaluate powers with negative exponents
• 8 Evaluate powers with negative or zero exponents
• 9 Multiply powers: integer bases
• 10 Divide powers: integer bases
• 11 Multiply and divide powers: integer bases
• 12 Power of a power: integer bases
• 13 Evaluate expressions using properties of exponents
• 14 Identify equivalent expressions involving exponents I
• 15 Identify equivalent expressions involving exponents II
• 16 Multiply powers: variable bases
• 17 Divide powers: variable bases
• 18 Multiply and divide powers: variable bases
• 19 Powers of a power: variable bases

D. Scientific notation

• 1 Convert between standard and scientific notation
• 2 Scientific notation on calculators
• 3 Compare numbers written in scientific notation
• 4 Add and subtract numbers written in scientific notation
• 5 Multiply numbers written in scientific notation
• 6 Divide numbers written in scientific notation

E. Square roots and cube roots

• 1 Square roots of perfect squares
• 2 Estimate positive square roots
• 3 Positive and negative square roots
• 4 Estimate positive and negative square roots
• 5 Relationship between squares and square roots
• 6 Solve equations using square roots
• 7 Cube roots of positive perfect cubes
• 8 Cube roots of positive and negative perfect cubes
• 9 Solve equations using cube roots
• 10 Estimate cube roots

F. Rational and irrational numbers

• 1 Identify rational and irrational square roots
• 2 Identify rational and irrational numbers
• 3 Classify numbers
• 4 Irrational numbers on number lines

G. Proportions

• 1 Solve proportions
• 2 Solve proportions: word problems
• 3 Estimate population size using proportions
• 4 Scale drawings: word problems
• 5 Scale drawings: scale factor word problems

H. Percents

• 1 Convert between percents, fractions, and decimals
• 2 Compare percents to fractions and decimals
• 3 Find what percent one number is of another
• 4 Find what percent one number is of another: word problems
• 5 Estimate percents of numbers
• 6 Percents of numbers and money amounts
• 7 Percents of numbers: word problems
• 8 Compare percents of numbers
• 9 Solve percent equations
• 10 Percent of change
• 11 Percent of change: word problems
• 12 Percent of change: find the original amount word problems

I. Consumer math

• 1 Price lists
• 2 Unit prices
• 3 Unit prices with unit conversions
• 4 Unit prices: find the total price
• 5 Percent of a number: tax, discount, and more
• 6 Find the percent: tax, discount, and more
• 7 Sale prices: find the original price
• 8 Multi-step problems with percents
• 9 Estimate tips
• 10 Simple interest
• 11 Compound interest

J. Units of measurement

• 1 Convert rates and measurements: customary units
• 2 Convert rates and measurements: metric units
• 3 Mixed customary units
• 4 Convert between customary and metric systems
• 5 Convert between Celsius and Fahrenheit

K. Expressions

• 1 Write variable expressions: one operation
• 2 Write variable expressions: two or three operations
• 3 Write variable expressions from diagrams
• 4 Write variable expressions: word problems
• 5 Evaluate one-variable expressions
• 6 Evaluate multi-variable expressions
• 7 Evaluate absolute value expressions
• 8 Evaluate radical expressions
• 9 Evaluate rational expressions
• 10 Identify terms and coefficients
• 11 Sort factors of variable expressions

L. Equivalent expressions

• 1 Properties of addition and multiplication
• 2 Multiply using the distributive property
• 3 Write equivalent expressions using properties
• 4 Add and subtract like terms
• 5 Add and subtract linear expressions
• 6 Factors of linear expressions
• 7 Identify equivalent linear expressions I
• 8 Identify equivalent linear expressions II
• 9 Identify equivalent linear expressions: word problems

M. One-variable equations

• 1 Which x satisfies an equation?
• 2 Write an equation from words
• 3 Model and solve equations using algebra tiles
• 4 Write and solve equations that represent diagrams
• 5 Properties of equality
• 6 Identify equivalent equations
• 7 Solve one-step equations
• 8 Solve two-step equations
• 9 Solve two-step equations: complete the solution
• 10 Solve one-step and two-step equations: word problems
• 11 Solve equations involving like terms
• 12 Solve equations with variables on both sides
• 13 Solve equations with variables on both sides: fractional coefficients
• 14 Solve equations with variables on both sides: word problems
• 15 Solve equations with the distributive property
• 16 Solve multi-step equations
• 17 Solve multi-step equations with fractional coefficients
• 18 Solve equations: mixed review
• 19 Solve multi-step equations: complete the solution
• 20 Find the number of solutions
• 21 Create equations with no solutions or infinitely many solutions

N. One-variable inequalities

• 1 Solutions to inequalities
• 2 Graph inequalities on number lines
• 3 Write inequalities from number lines
• 4 Solve one-step inequalities
• 5 Graph solutions to one-step inequalities
• 6 Solve two-step inequalities
• 7 Graph solutions to two-step inequalities
• 8 Solve multi-step inequalities
• 9 Graph solutions to multi-step inequalities
• 10 Solve inequalities with integers: variables on both sides
• 11 Solve inequalities with decimals: variables on both sides

O. Coordinate plane

• 1 Coordinate plane review
• 2 Quadrants and axes
• 3 Follow directions on a coordinate plane
• 4 Find the distance between two points

P. Lines and angles

• 1 Identify complementary, supplementary, vertical, adjacent, and congruent angles
• 2 Find measures of complementary, supplementary, vertical, and adjacent angles
• 3 Write and solve equations using angle relationships
• 4 Identify alternate interior and alternate exterior angles
• 5 Transversals of parallel lines: name angle pairs
• 6 Transversals of parallel lines: find angle measures
• 7 Transversals of parallel lines: solve for x
• 8 Find lengths and measures of bisected line segments and angles

Q. Two-dimensional figures

• 1 Identify and classify polygons
• 2 Classify triangles
• 3 Identify trapezoids
• 4 Classify quadrilaterals I
• 5 Classify quadrilaterals II
• 6 Graph triangles and quadrilaterals
• 7 Find missing angles in triangles
• 8 Find missing angles in triangles using ratios
• 9 Triangle Angle-Sum Theorem
• 10 Find missing angles in quadrilaterals I
• 11 Find missing angles in quadrilaterals II
• 12 Exterior Angle Theorem
• 13 Interior angles of polygons
• 14 Parts of a circle

R. Transformations and congruence

• 1 Identify reflections, rotations, and translations
• 2 Describe a sequence of transformations
• 3 Translations: graph the image
• 4 Translations: find the coordinates
• 5 Translations: write the rule
• 6 Reflections over the x- and y-axes: graph the image
• 7 Reflections over the x- and y-axes: find the coordinates
• 8 Reflections: graph the image
• 9 Reflections: find the coordinates
• 10 Rotations: graph the image
• 11 Rotations: find the coordinates
• 12 Reflections and rotations: write the rule
• 13 Describe transformations
• 14 Sequences of congruence transformations: graph the image
• 15 Sequences of congruence transformations: choose the sequence
• 16 Identify congruent figures
• 17 Congruence statements and corresponding parts
• 18 Determine if two figures are congruent: justify your answer
• 19 Side lengths and angle measures of congruent figures

S. Transformations and similarity

• 1 Similar and congruent figures
• 2 Dilations: graph the image
• 3 Dilations: find the coordinates
• 4 Dilations: find the scale factor
• 5 Identify similar triangles
• 6 Angle-angle criterion for similar triangles
• 7 Side lengths and angle measures of similar triangles
• 8 Side lengths and angle measures of similar figures
• 9 Similar triangles and indirect measurement
• 10 Find missing side lengths in proportional triangles

T. Pythagorean theorem

• 1 Pythagorean theorem: find the length of the hypotenuse
• 2 Pythagorean theorem: find the missing leg length
• 3 Pythagorean theorem: find the missing leg or hypotenuse length
• 4 Pythagorean theorem: find the perimeter
• 5 Pythagorean theorem: word problems
• 6 Converse of the Pythagorean theorem: is it a right triangle?

U. Three-dimensional figures

• 1 Parts of three-dimensional figures
• 2 Nets of three-dimensional figures
• 3 Front, side, and top view
• 4 Similar solids

V. Perimeter and area

• 1 Perimeter
• 3 Area and perimeter: word problems
• 4 Area and circumference of circles
• 5 Circles: word problems
• 6 Area and perimeter of semicircles and quarter circles
• 7 Area between two shapes
• 8 Perimeter and area: changes in scale

W. Surface area and volume

• 1 Volume of cubes, prisms, and pyramids
• 2 Surface area of cubes, prisms, and pyramids
• 3 Volume of cylinders
• 4 Volume of cones
• 5 Surface area of cylinders
• 6 Surface area of cones
• 7 Volume of spheres
• 8 Surface area of spheres
• 9 Volume and surface area of similar solids

X. Proportional relationships

• 1 Find the constant of proportionality from a table
• 2 Write equations for proportional relationships from tables
• 3 Identify proportional relationships by graphing
• 4 Find the constant of proportionality from a graph
• 5 Write equations for proportional relationships from graphs
• 6 Identify proportional relationships from graphs and equations
• 7 Identify proportional relationships from tables
• 8 Identify proportional relationships: word problems
• 9 Graph proportional relationships and find the slope
• 10 Interpret graphs of proportional relationships
• 11 Write and solve equations for proportional relationships
• 12 Compare proportional relationships represented in different ways

Y. Direct variation

• 1 Find the constant of variation
• 2 Identify direct variation
• 3 Write direct variation equations
• 4 Write and solve direct variation equations
• 1 Find the slope from a graph
• 2 Find the slope from two points
• 3 Find the slope from a table
• 4 Find a missing coordinate using slope
• 5 Graph a line using slope

AA. Linear equations

• 1 Is (x, y) a solution to the linear equation?
• 2 Relate the graph of an equation to its solutions
• 3 Slope-intercept form: find the slope and y-intercept
• 4 Graph a line from an equation in slope-intercept form
• 5 Graph a line from an equation in point-slope form
• 6 Write a linear equation from a slope and y-intercept
• 7 Write a linear equation from a graph
• 8 Write a linear equation from a slope and a point
• 9 Write a linear equation from two points
• 10 Convert a linear equation in standard form to slope-intercept form
• 11 Graph a line from an equation in standard form
• 12 Graph a horizontal or vertical line
• 13 Equations of horizontal and vertical lines
• 14 Slopes of parallel and perpendicular lines

BB. Function concepts

• 1 Identify functions
• 2 Identify functions: graphs
• 3 Identify independent and dependent variables
• 4 Find values using function graphs
• 5 Complete a table for a function graph
• 6 Domain and range of functions

CC. Linear functions

• 1 Evaluate a linear function
• 2 Complete a table for a linear function
• 3 Complete a table and graph a linear function
• 4 Interpret points on the graph of a linear function
• 5 Rate of change of a linear function: graphs
• 6 Interpret the slope and y-intercept of a linear function
• 7 Write a linear function from a table
• 8 Compare linear functions: graphs and equations
• 9 Compare linear functions: tables, graphs, and equations
• 10 Write linear functions: word problems
• 11 Evaluate a linear function: word problems

DD. Nonlinear functions

• 1 Identify linear and nonlinear functions: graphs and equations
• 2 Identify linear and nonlinear functions: tables
• 3 Is (x, y) a solution to the nonlinear equation?
• 4 Evaluate a nonlinear function

EE. Interpret functions

• 1 Rate of change: tables
• 2 Rate of change: graphs
• 3 Identify graphs: word problems

FF. Sequences

• 1 Identify arithmetic and geometric sequences
• 2 Arithmetic sequences
• 3 Geometric sequences
• 4 Sequences: mixed review
• 5 Sequences: word problems
• 6 Evaluate variable expressions for sequences
• 7 Write variable expressions for arithmetic sequences

GG. Systems of equations

• 1 Is (x, y) a solution to the system of equations?
• 2 Solve a system of equations by graphing
• 3 Solve a system of equations by graphing: word problems
• 4 Find the number of solutions to a system of equations by graphing
• 5 Find the number of solutions to a system of equations
• 6 Classify a system of equations by graphing
• 7 Classify a system of equations
• 8 Solve a system of equations using substitution
• 9 Solve a system of equations using substitution: word problems
• 10 Solve a system of equations using elimination
• 11 Solve a system of equations using elimination: word problems
• 12 Solve a system of equations using any method
• 13 Solve a system of equations using any method: word problems

HH. One-variable statistics

• 1 Calculate mean, median, mode, and range
• 2 Interpret charts and graphs to find mean, median, mode, and range
• 3 Mean, median, mode, and range: find the missing number
• 4 Changes in mean, median, mode, and range
• 5 Calculate mean absolute deviation
• 6 Calculate quartiles and interquartile range
• 7 Box plots
• 8 Identify an outlier
• 9 Identify an outlier and describe the effect of removing it

II. Two-variable statistics

• 1 Interpret line graphs
• 2 Create line graphs
• 3 Interpret scatter plots
• 4 Create scatter plots
• 5 Identify trends with scatter plots
• 6 Make predictions with scatter plots
• 7 Outliers in scatter plots
• 8 Identify lines of best fit
• 9 Write equations for lines of best fit
• 10 Interpret lines of best fit: word problems
• 11 Identify representative, random, and biased samples

JJ. Probability

• 1 Probability of simple events
• 2 Probability of opposite, mutually exclusive, and overlapping events
• 3 Experimental probability
• 4 Find probabilities using two-way frequency tables
• 5 Make predictions
• 6 Compound events: find the number of outcomes
• 7 Compound events: find the number of sums
• 8 Identify independent and dependent events
• 9 Probability of independent and dependent events
• 10 Counting principle

Eighth grade lessons

These lessons help you brush up on important math topics and prepare you to dive into skill practice!

• Additive inverses
• Adding and subtracting integers
• Multiplying and dividing integers

Rational and irrational numbers

• Rational numbers
• Adding and subtracting rational numbers
• Multiplying and dividing rational numbers
• Irrational numbers

Exponents, scientific notation, and roots

• Properties of exponents
• Scientific notation
• Adding and subtracting numbers in scientific notation
• Multiplying and dividing numbers in scientific notation
• Square roots
• Proportional relationships
• Constant of proportionality
• Simple interest
• Percent change

Two-dimensional figures

• Parts of a circle
• Area of circles
• Circumference of circles
• Complementary angles
• Supplementary angles
• Adjacent angles
• Vertical angles
• Transversals of parallel lines
• Interior angles of triangles
• Exterior angles of triangles
• The Pythagorean theorem
• The converse of the Pythagorean theorem
• Distance formula

Transformations, congruence, and similarity

• Transformations
• Translations
• Reflections
• Similar triangles

Three-dimensional figures

• Surface area formulas
• Volume formulas
• Volume of prisms
• Volume of cylinders
• Volume of pyramids
• Volume of cones
• Volume of spheres

Expressions

• Writing algebraic expressions
• Evaluating expressions
• Simplifying expressions
• The distributive property
• Expanding expressions
• Factoring expressions

Equations and inequalities

• Solving equations
• Multi-step equations
• Equations with infinitely many or no solutions
• Solving inequalities
• Linear equations
• Slope-intercept form
• Point-slope form
• Standard form of linear equations
• Equations of parallel and perpendicular lines
• Systems of equations
• Relations and functions
• Independent and dependent variables
• Linear functions
• Domain and range

Data and graphs

• Box and whisker plots
• Scatter plots
• Correlation
• Line of best fit
• Two-way tables

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8th Grade Math : Geometry

Study concepts, example questions & explanations for 8th grade math, all 8th grade math resources, example questions, example question #1 : geometry.

First, we need to define some key terms:

Parallel Lines : Parallel lines are lines that will never intersect with each other. 

Transversal Line : A transversal line is a line that crosses two parallel lines.

It is important to know that transversal lines create angle relationships:

• Vertical angles are congruent
• Corresponding angles are congruent
• Alternate interior angles are congruent
• Alternate exterior angles are congruent

Calculate the length of the missing side of the provided triangle. Round the answer to the nearest whole number.

We can use the Pythagorean Theorem to help us solve this problem.

The Pythagorean Theorem states that for right triangles, the square of the hypotenuse is equal to the sum of the square of the other two sides. In other terms:

We can use the formula and substitute the known side lengths from the problem to solve for the missing side length:

Example Question #3 : Geometry

Calculate the volume of the cone provided. Round the answer to the nearest hundredth. 

In order to solve this problem, we need to recall the formula used to calculate the volume of a cone:

Now that we have this formula, we can substitute in the given values and solve:

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Free Printable Math Word Problems Worksheets for 8th Grade

Math Word Problems: Discover a vast collection of free printable worksheets for Grade 8 students, created by Quizizz. Enhance your students' problem-solving skills and mathematical understanding with these comprehensive resources.

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Explore printable Math Word Problems worksheets for 8th Grade

Math Word Problems worksheets for Grade 8 are an essential resource for teachers looking to challenge their students and help them develop critical thinking and problem-solving skills. These worksheets cover a wide range of topics, including algebra, geometry, and statistics, ensuring that students have a solid foundation in math. By incorporating real-life scenarios and engaging themes, these worksheets not only make math more relatable but also help students see the practical applications of the concepts they are learning. Teachers can easily incorporate these worksheets into their lesson plans, using them as supplementary material, homework assignments, or even as assessment tools. With a variety of difficulty levels and question types, Math Word Problems worksheets for Grade 8 cater to the diverse needs of students and ensure that they are well-prepared for future math courses.

Quizizz is a fantastic platform that offers not only Math Word Problems worksheets for Grade 8 but also a plethora of other resources for teachers to enhance their students' learning experience. With Quizizz, teachers can create interactive quizzes, polls, and presentations to engage students and make learning fun. The platform also allows teachers to track their students' progress and identify areas where they may need additional support. In addition to Math Word Problems worksheets for Grade 8, Quizizz offers resources for other subjects and grade levels, making it a one-stop-shop for all your educational needs. The user-friendly interface and customizable features make Quizizz an invaluable tool for teachers looking to incorporate technology into their classrooms and provide their students with a dynamic and interactive learning experience.

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New Brunswick Mathematics Curriculum

Students learn the basic concepts of algebra, geometry, and graphing before entering high school and learning more complicated topics. It is crucial that students completely grasp these concepts before going on to harder topics like trigonometry or calculus. With our on-demand videos students have the opportunity to go over math problems with a math teacher who knows how to break it down in an easily digestible format.

• Teachers go over definitions along with multiple problems for the skills so that students fully grasp the concepts.
• Skills available for statistics, time, ratios, and other eighth grade skills.
• Students learn how to use the scracthpad to better understand how to tackle the problems.

Grade 8 Algebra Word Problems

These lessons cover some examples and solutions for algebra word problems that you will commonly encounter in grade 8.

Related Pages More Math Word Problems Algebra Word Problems

How to write algebra word problems into systems of linear equations and solve systems of linear equations using elimination and substitution methods?

Grade 8 Algebra Word Problems How to solve algebra word problems using systems of linear equations?

Example: Devon is going to make 3 shelves for her father. He has a piece of lumber 12 feet long. She wants the top shelf to be half a foot shorter than the middle shelf, and the bottom shelf to be half a foot shorter than twice the length of the top shelf. How long will each shelf be if she uses the entire 12 feet of wood?

Grade 8 Algebra Word Problems - Line Segments

Example: If JK = 7x + 9, JL = 114 and KL = 9x + 9. Find KL.

Grade 8 number word problems - common core How to write word problems into systems of linear equations and solve systems of linear equations using elimination and substitution methods?

Example 1: The sum of two numbers is 361 and the difference between the two numbers is 173. What are the two numbers?

Example 2: There are 356 Grade 8 students at Euclid’s Middle School. Thirty-four more than four times the number of girls is equal to half the number of boys. How many boys are in Grade 8 at Euclid’s Middle School? How many girls?

Example 3: A family member has some five-dollar bills and one-dollar bills in their wallet. Altogether she has 18 bills and a total of $62. How many of each bill does she have?

Example 1: A friend bought 2 boxes of pencils and 8 notebooks for school and it cost him $11. He went back to the store the same day to buy school supplies for his younger brother. He spent $11.25 on 3 boxes of pencils and 5 notebooks. How much would notebooks cost?

• A farm raises cows and chickens. The farmer has a total of 42 animals. One day he counts the legs of all of his animals and realizes he has a total of 114. How many cows does the farmer have? How many chickens?
• The length of a rectangle is 4 times the width. The perimeter of the rectangle is 45 inches. What is the area of the rectangle?
• The sum of the measures of angles x and y is 127". If the measure of angle x is 34° more than half the measure of angle y, what is the measure of each angle?

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Solving more geometric problems

12.6 Solving more geometric problems

When solving geometry problems involving lines, you need to use some facts about quadrilaterals and triangles.

The sum of the interior angles of a triangle is \(180^\circ\). The exterior angle of a triangle is equal to the sum of the interior opposite angles. exterior angle of a triangle the angle formed outside a triangle when one of its sides is produced The interior angles of a quadrilateral add up to \(360^{\circ}\). \(\hat{P}+\hat{Q}+\hat{R}+\hat{S}= 360^{\circ}\) interior angle an angle that lies inside a shape

Worked example 12.9: Identifying angles on parallel lines

Calculate the size of angles \(a\), \(b\), \(c\) and \(d\).

Examine the parallel lines. Identify how many transversals there are.

Notice that there are two traversals: \(SQ\) and \(UR\). Now look for the angles that form F or Z pattern along each transversal.

• Transversal \(UR\): We can see that \(b\) is a corresponding angle to \(48^{\circ}\), making an F pattern. There are no alternate angles formed by this transversal.
• Transversal \(SQ\): We can see that \(a\) is alternate to \(c\), making a Z pattern. There are no corresponding angles formed by this transversal.

Look for any adjacent supplementary angles that can help you find one of the angles.

Angle \(c\) is an adjacent supplementary angle to 154°.

Write out your solution.

Worked example 12.10: using angles on parallel lines.

Calculate the size of \(x\).

The two given angles lie between parallel lines, and are corresponding angles, therefore they are equal. We will use this fact to form an equation to help us solve for \(x\).

Worked example 12.11: using angles on parallel lines.

Notice that there is only one transversal in this diagram. The two given angles make a C pattern.

The two angles form co-interior angles, which add up to \(180^{\circ}\), because the lines are parallel. So we can write out an equation using this information.

Worked example 12.12: Solving geometric problems involving properties of triangles

Calculate the sizes of \(\hat{1}\) to \(\hat{6}\).

Examine the diagram carefully. Look for the parallel lines and all transversals.

Notice that there are two transversals; \(AD\) and \(OB\):

• Looking at \(AD\), notice that \(\hat{2}\) and \(\hat{6}\) are alternate angles and therefore equal. We also see that \(\hat{1}\) and \(\hat{6}\) are corresponding angles and therefore equal.
• Looking at \(OB\), we notice that \(\hat{4}\) and \(\hat{5}\) are alternate angles.

Look for all the adjacent angles on a straight line.

You may need to use the fact they are supplementary later in your calculation. How many supplementary angles can you find?

The supplementary angles are:

Look for vertically opposite angles.

How many can you find? (\(\hat{1}\) is vertically opposite \(\hat{2}\)).

Hint: Remember that in an isoceles triangle, two sides are the same length.

Because \(\triangle ABO\) is an isoceles triangle:

Worked example 12.13: Solving geometric problems involving properties of triangles and quadrilaterals

\(ABCD)\ is a parallelogram. Calculate the sizes of \(A\hat{B}D\), \(A\hat{D}B\), \(\hat{C}\) and \(C\hat{B}D\).

Use the properties of the parallelogram \(ABCD\) to expand the given information.

Both pairs of opposite angles are equal in a parallelogram, therefore we can say that \(\hat{A}=\hat{C}\).

Look for any alternate or corresponding angles for both sets of parallel lines. We also want to make of a note of any co-interior angles formed by the parallel lines. These supplementary angles may also be useful in helping us to find the missing angles.

We are given two sets of parallel lines: \(AD \parallel BC\) and \(DC \parallel AB\).

• \(DC \parallel AB\): Looking at transversal \(BD\), we notice that \(20^{\circ}\) and \(\hat{DBA}\) are alternate angles formed by these parallel lines, and therefore are equal.
• \(AD \parallel BC\): Looking at transversal \(BD\), we notice that \(\hat{ADB}\) and \(\hat{DBC}\) are alternate angles formed by these parallel lines, and therefore are equal.

Use the properties of triangles to find angles.

We are also given two triangles: \(\triangle{ABD}\) and \(\triangle{DBC}\). We can use the fact that the sum of \(\angle\)s of a \(\triangle\) = \(180^\circ\) to help find remaining angles. 

\(RSTU\) is a trapezium. Calculate the sizes of \(\hat{T}\) and \(\hat{R}\).

\(JKLM\) is a rhombus. Calculate the sizes of of \(J\hat{M}L\), \(\hat{M}_2\) and \(\hat{K}_1\).

Hint: Remember, \(JKLM\) is a rhombus, therefore all four sides of \(JKLM\) are equal.

In the diagram, \(AB \parallel CD\). Calculate the sizes of \(F\hat{H}G, \hat{F},\hat{C}\) and \(\hat{D}\). Give reasons for your answers.

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How Math Autobiographies Build Student Confidence

Incorporating a writing task into lessons can help students reflect on their work and see their skills in a new light.

As educators, we understand the crucial role of fostering confidence and a strong math identity in our students. We aim for each student to feel empowered and to see themselves as capable mathematicians every day. One effective way to nurture this mindset is by encouraging students to write a math autobiography. This reflective exercise invites them to consider their entire mathematical journey, which helps build confidence and reinforce the belief that they are inherently skilled at math.

Writing a math autobiography allows students to connect with each other through their mathematical experiences, identify their learning preferences (though those may change), and set future goals. It also highlights the times when they’ve stepped out of their comfort zone and found joy in the challenges they faced. Take a fifth grader, who discovered that “the more challenging math got, the more I loved it.” This realization helped her embrace challenges, because she knew she could enjoy them. 

Similarly, another fifth grader reflected on his experience learning math during the pandemic. He found that virtual learning strengthened his perseverance, and he realized the power of his own resilience. By integrating reading, writing, and talking, this project guides students to develop a mathematical mindset. 

Would Students Describe Themselves as Mathematicians? 

Most students wouldn’t know how to respond, but we know mathematicians each have their own unique and beautiful math history. We can guide students by helping them find what they enjoy as a mathematician, how they hope to be perceived as a mathematician, and what concepts they’re passionate about. In our experience, students uncovered their math identity by using mentor texts and emotion graphs .

Mentor texts provide a mathematician’s perspective while allowing students to reflect on their own math journey. We chose to share mentor texts centered around the theme of perseverance. When we read Nothing Stopped Sophie , by Cheryl Bardoe, a fourth-grade mathematician stated, ”I forget that every person has trouble with math sometimes.” 

We used emotion graphs to help students share their stories and experiences, which allowed them to realize what has shaped their mindset. This visual brainstorming task was the foundation for their math autobiography. As students thought about their math feelings throughout the years, memories flooded into their heads. 

For our upper elementary students, we decided to use three different emojis on the y axis (happy, indifferent, and sad). The x axis listed the time period of their life. Most students had a “before school” category and then listed each grade level individually. 

Planning on Paper and Identifying Relatable Themes 

With these connections made, students were ready to plan their math autobiographies. Each student got six sticky notes to jot ideas related to their own history. When students read biographies, they look for certain elements (dates, challenges or obstacles, achievements, memorable moments, firsts and lasts, and connections or relationships).

They consider their own math history with each of these lenses, which gives them a collection of ideas to get writing. When students place each sticky note on a sheet of plain white paper, they’re able to view all of their ideas at one time and begin to plan their own autobiography. As they write, they can return to their sticky notes over and over again, weaving in each component. Students might plan ahead by adding a star to certain ideas that they want to include. They might also use check marks to keep track of what they’ve included so far. 

In addition, having the sheet with these sticky notes at student workspaces will help teachers engage in writing conferences to help students get started when they’re stuck (and also help students rehearse ideas with partners before writing).

As students began writing, we noticed that each one crafted their own theme. One student reflected on his confidence in math when he wrote, “When I was in kindergarten, that’s when math started to play a role in my life” and “I realized it was not good to be overconfident in my math skills.”

Another student highlighted her perseverance in her math autobiography, saying, “I knew after trying and working hard, I could do it!” A student chose to focus his writing around connecting mathematics to world experiences when he wrote, “Math, I found, was everywhere—in Sesame Street , on cereal boxes, and in chess, my favorite board game—and so I grew as a mathematician.”

Instructional Strategies That Empower Mathematicians

Having the texts out while students are writing is key. Students quickly pointed out that the mentor texts they were reading had illustrations and pictures, so they included pictures in their own math autobiographies. Students wrote captions to convey the theme of their photographs. 

When describing a candy photo he had pasted on his autobiography, a fifth grader wrote, “Mathematicians sort objects. On Halloween, me and my friends gave ‘values’ to every candy to make trading easier.” Mathematicians became motivated to teach others through their own math story. As a result, they became more confident in their math identity.

Showcase Student Projects in Different Ways 

Some students created storybooks, others designed scrapbook pages, and still others wrote by hand, typed, or recorded their voice. Projects included real photographs, graphs, and math symbols in their design. A gallery walk displayed the math autobiographies so that everyone could move around to see the projects. 

In a follow-up discussion, consider asking students: 

• What’s the same about your work and someone else’s work? What is different? 
• What kind of mathematicians are we? Can you finish this statement, “We’re the kind of mathematicians who…”?
• What can we do with math? How would you finish this sentence, “With math, we can…”? (Do this a few times in different ways.) 

Embedding this writing project into any math classroom can help build confidence in students and promote reflection. Elementary mathematicians will craft a successful autobiography when they have time to reflect on their feelings around math, brainstorm memorable moments, and connect with other mathematicians. 

Reflecting on their current feelings about their identity, the more they reflect, the more they’ll grow in and outside of the classroom.

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Why Britain Just Ended 14 Years of Conservative Rule

Last week, the center-left labour party won the british general election in a landslide..

Hosted by Natalie Kitroeff

Featuring Mark Landler

Produced by Rob Szypko ,  Nina Feldman and Will Reid

Edited by Brendan Klinkenberg

With Paige Cowett

Original music by Dan Powell ,  Diane Wong and Marion Lozano

Engineered by Alyssa Moxley

Listen and follow The Daily Apple Podcasts | Spotify | Amazon Music | YouTube

For more than a decade, Britain has been governed by the Conservative Party, which pushed its politics to the right, embracing smaller government and Brexit. Last week, that era officially came to an end.

Mark Landler, the London bureau chief for The Times, explains why British voters rejected the Conservatives and what their defeat means in a world where populism is on the rise.

On today’s episode

Mark Landler , the London bureau chief for The New York Times.

Background reading

Five takeaways from the British general election.

The Conservatives have run Britain for 14 years. How have things changed in that time?

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Michael Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Maddy Masiello, Isabella Anderson, Nina Lassam and Nick Pitman.

Natalie Kitroeff is the Mexico City bureau chief for The Times, leading coverage of Mexico, Central America and the Caribbean. More about Natalie Kitroeff

Mark Landler is the London bureau chief of The Times, covering the United Kingdom, as well as American foreign policy in Europe, Asia and the Middle East. He has been a journalist for more than three decades. More about Mark Landler

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Unit 2: Solving equations with one unknown

About this unit.

Let's take your equation-solving skills to the next level! You'll learn all the strategies you need to confidently tackle problems with variables on both sides, parentheses, and more. Plus, you'll get plenty of practice with word problems to really solidify your understanding.

Equations with variables on both sides

• Intro to equations with variables on both sides (Opens a modal)
• Equations with variables on both sides: 20-7x=6x-6 (Opens a modal)
• Equation with variables on both sides: fractions (Opens a modal)
• Equation with the variable in the denominator (Opens a modal)
• Equations with variables on both sides Get 3 of 4 questions to level up!
• Equations with variables on both sides: decimals & fractions Get 3 of 4 questions to level up!

Equations with parentheses

• Equations with parentheses (Opens a modal)
• Multi-step equations review (Opens a modal)
• Equations with parentheses Get 3 of 4 questions to level up!
• Equations with parentheses: decimals & fractions Get 3 of 4 questions to level up!

Number of solutions to equations

• Number of solutions to equations (Opens a modal)
• Worked example: number of solutions to equations (Opens a modal)
• Creating an equation with no solutions (Opens a modal)
• Creating an equation with infinitely many solutions (Opens a modal)
• Number of solutions to equations Get 3 of 4 questions to level up!
• Number of solutions to equations challenge Get 3 of 4 questions to level up!

Equations word problems

• Sums of consecutive integers (Opens a modal)
• Sum of integers challenge (Opens a modal)
• Solving equations with one unknown: FAQ (Opens a modal)
• Sums of consecutive integers Get 3 of 4 questions to level up!