math problem solving for grade 10

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Grade 10 math questions and solutions.

The Smarter Balanced Assessment Consortium (SBAC) is a standardized test that includes a variety of new technology-enhanced questions.

Some of them are Multiple choice-single correct responses, Multiple choice-multiple correct responses, Matching Tables, Drag and Drop, Hot text, Table Fill in, Graphing, Equation/numeric, Extended constructed response, Short answer, and many more.

This page contains several sample questions along with practice test links for Grade 10 Math that gives you an idea of questions that your students are likely to see on the test. After each sample question, an answer explanation follows. The explanation includes essential aspects of the task that you may need to consider for the skills, processes, and information your students need to know.

Domain: Grade 10 >> Number and Quantity – The Real Number System

Sample Question: Evaluate 9 150/300

Answer Explanation: 9 150/300 = 9 1/2 = square root of 9 = 3. In a problem with a rational exponent, the numerator tells you the power, and the denominator the root. However, in this problem the exponent can be reduced, so we should reduce that first. The exponent 150/300 = 1/2. So the problem becomes 9 to the 1/2 power. The denominator is 2 so we take the square root of 9 which is 3. The numerator is 1 so we raise 3 to the 1st power and the answer is 3.

Standards: HSN.RN.A.1

Click here to practice: Number and Quantity – The Real Number System Questions on Grade 10 Math

Domain: Grade 10 >> Number and Quantity – Quantities

Sample Question: Rewrite x 1/2 in radical form.

Answer Explanation: In a problem with a rational exponent, the numerator tells you the power, and the denominator the root. Since the problem is, x 1/2 , the denominator is 2 indicating we should take a square root and the numerator is 1 so we would raise that to the first power or there will be no exponent since an exponent of 1 is rarely used. That makes the answer the square root of x, written as √x.

Click here to practice: Grade 10 Number and Quantity – Quantities Questions

Domain: Grade 10 >> Number and Quantity – The Complex Number System

Sample Question: Simplify completely i(7−i)

Answer Explanation: i(7−i)=i*7−i*i=7i−i 2 =7i−(−1)=7i+1=1+7i

Start by using the distributive method. Now simplify −i 2 =1 by definition. Now rearrange and put the real part first and the imaginary part last so that it looks like this a+bi.

Standards: HSN.CN.A.2

Click here to practice: Grade 10 Math Number and Quantity – The Complex Number System Questions

Domain: Grade 10 >> Number and Quantity – Vector & Matrix Quantities

Sample Question: A vector in standard form has components . What is the initial point?

• Not enough information given

Answer Explanation: Since the vector is in standard position, we know that the initial point is (0, 0) or the origin.

Standards: HSN.VM.A.2

Click here to practice: Number and Quantity – Vector & Matrix Quantities Questions for Grade 10 Math

Domain: Grade 10 >> Algebra – Seeing Structure in Expressions

Sample Question: Which expression is equivalent to 9x 2 – 16y 2 ?

• (3x – 4y) (3x – 4y)
• (3x + 4y) (3x + 4y)
• (3x + 4y) (3x – 4y)
• (3x – 4y) 2

Answer Explanation: Student must recognize the expression is the difference of two perfect squares

Standards: HSA.SSE.A.2

Click here to practice: Algebra – Seeing Structure in Expressions Questions for Grade 10 Math

Domain: Grade 10 >> Algebra – Arithmetic with Polynomials & Rational Expressions

Sample Question: Evaluate f(x)=−a 3 +6a−7 at a = – 1 and state the remainder.

Answer Explanation: student must substitute – 1 into the function as follows −(−1) 3 +6(−1)−7=−12 and find the value to get the remainder

Standards: HSA.APR.B.2

Click here to practice: Algebra – Arithmetic with Polynomials & Rational Expressions Questions for Grade 10 Math

Domain: Grade 10 >> Algebra – Creating Equations

Sample Question: The ratio of staff to guests at the gala was 3 to 5. There were a total of 576 people in the ballroom. How many guests were at the gala?

Answer Explanation: Setup a proportion of guests to the total number of people, 8/5 = x/576. Solve by cross multiplying. 8x = 2880. Divide both sides by 8. So x=360.

Standards: HSA.CED.A.3

Click here to practice: Algebra – Creating Equations Questions for Grade 10 Math

Domain: Grade 10 >> Algebra – Reasoning with Equations & Inequalities

Sample Question: Solve the quadratic x 2 +10x=−25.

Answer Explanation: This problem can be easily solved by rearranging the equation so that it is solved for zero and then factoring as shown:

x 2 +10x=−25

x 2 +10x+25=0

(x+5)(x+5)=0

Since both factors are exactly the same, you will only have one solution to this problem.

Standards: HSA.REI.B.4

Click here to practice: Algebra – Reasoning with Equations & Inequalities Questions for Grade 10 Math

Domain: Grade 10 >> Functions – Interpreting Functions

Sample Question: Which graph could represent the graph of f(x)=sin(x)?

Answer Explanation: The sin function always graphs to look like a wave. The only one that could be the sin function is D.

Standards: HSF.IF.C.7

Click here to practice: Functions – Interpreting Functions Questions for Grade 10 Math

Domain: Grade 10 >> Functions – Building Functions

Sample Question: Describe how the graph of g(x)=x 3 – 5 can be obtained by shifting f(x) = x 3 + 2.

• Shift right 7 units
• Shift left 7 units
• Shift up 7 units
• Shift down 7 units

Answer Explanation: The only thing that changed in the two equations is the y-intercept which controls the vertical shift (up or down). To get the graph of g(x) by shifting the graph of f(x), you would shift f(x) down 7 units to change from +2 to -5.

Standards: HSF.BF.B.3

Click here to practice: Functions – Building Functions Questions for Grade 10 Math

Sample Question: Solve 3 x =12 using logarithmic form.

• x = ln12/ln3
• None of these

Answer Explanation: Solve using logs as follows 3 x =12 x=log(base 3) 12 x=ln12/ln3

Standards: HSF.LE.A.4

Domain: Grade 10 >> Functions – Trigonometric Functions

Sample Question: In the unit circle, one can see that tan(5π/4)=1 . What is the value of cos(5π/4)?

Standards: HSF.TF.A.2

Click here to practice: Functions – Trigonometric Functions Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Congruence

Sample Question: What would be the coordinates of point S after applying the following rule: (x+3, y -2)?

Answer Explanation: Answer: B Explanation: The transformation rule that is give is to translate the point 3 units to the right and 2 units down as shown by the following diagram:

Click here to practice: Geometry – Congruence Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Similarity, Right Triangles, & Trigonometry

Sample Question: By what property can the angles BAX and TSX be found to be congruent?

• Corresponding angles
• Vertical angles
• Alternate interior angles
• Congruent angles

Answer Explanation: Answer: A

Although they are congruent angles, the question is asking for the property. Since they are in corresponding locations with the transversal (AX) the correct answer is A

Standards: HSG.SRT.A.3

Click here to practice: Geometry – Similarity, Right Triangles, & Trigonometry Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Circles

Sample Question: What is the translation rule and the scale factor of the dilation as Circle F→Circle F′ ?

• (x,y)→1/4(x,y+10)
• (x,y)→4(x,y+10)
• (x,y)→1/4(x+10,y)
• (x,y)→1/4(x,y−10)

Answer Explanation: The original circle F has its center at the point (−5,−6) with a radius of 4 units. The translated/dilated circle F’ has its center at the point (−5,4) with a radius of 1 units. This means the center was translated up 10 units. As a transformation, this translation is written as (x,y)→(x,y+10). Circle F was also dilated by a factor of 1/4 because the radius was reduced from 4 units to 1 units. As a transformation, this dilation is written as (x,y)→1/4(x,y). Putting the translation and dilation together, the rule is (x,y)→1/4(x,y+10).

Click here to practice: Geometry – Circles Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Expressing Geometric Properties with Equations

Sample Question: What value on the number line in the figure below divides segment EF into two parts having a ratio of their lengths of 3:1?

Answer Explanation: Point E is at -7 on the number line in the figure, and pointF is at 1. Thus, the length of segment EF is 8. To divide the segment into two parts with a ratio of their lengths of 3:1, change the ratio to 3x:1x to allow variation in the location on the number line. Next, set the sum of the two parts equal to 8 and solve for x. 3x+1x=8;4x=8;x=2.Now, that you know that x=2, find 3x, which equals 6. Find the value on the number line by adding 6 to the position of point E. −7+6=−1.The value on the number line that divides segment EF in a ratio of 3:1 is -1.

Standards: HSG.GPE.B.6

Click here to practice: Geometry – Expressing Geometric Properties with Equations Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Geometric Measurement & Dimension

Sample Question: What is the volume of the prism shown below?

Answer Explanation: Use the formula for volume of a pyramid:

V=1/2.a.c.h

In this case the length is 15cm, the base is 10 cm in length, and the height is 9 cm. Therefore :

V=1/2.15.10.9=675cm 3

Standards: HSG.GMD.A.3

Click here to practice: Geometry – Geometric Measurement & Dimension Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Modeling with Geometry

Sample Question: A company ships spherical paperweights in cubic boxes. The circumference of the paperweight is 9π cm. If the box fits the sphere exactly with the sides of the sphere touching the box, what is the volume of the smallest box the company can use for shipping.

• 1009 π cm 3

Standards: HSG.MG.A.3

Click here to practice: Geometry – Modeling with Geometry Questions for Grade 10 Math

Domain: Grade 10 >> Statistics & Probability – Interpreting Categorical & Quantitative Data

Sample Question: Given the scatter plot below, what type of function expresses the correlation between the two variables?

• Exponential

Answer Explanation: Notice that the trend of the graph (in red) between the data points forms a line.

Standards: HSS.ID.A.4

Click here to practice: Statistics & Probability – Interpreting Categorical & Quantitative Data Questions for Grade 10 Math

Domain: Grade 10 >> Statistics & Probability – Making Inferences & Justifying Conclusions

Sample Question: In a research project about pet behavior, a random sample of 400 cats was chosen. The study showed that 60% of the cats preferred to sleep inside the house. Chicken was the favorite food for 35% of those cats. The study also showed that 85% of the cats that preferred to sleep outside the house had a different favorite dish. How many cats in the sample liked chicken the best and preferred to sleep inside?

Answer Explanation: If the sample has 400 cats and 60% of the cats preferred to sleep inside, then 400.0.60=240 cats preferred to sleep inside and 160 cats preferred to sleep outside. Next, if the favorite dish of 35% of those cats that preferred to sleep inside was chicken, then, 240.0.35=84 cats in the sample preferred to sleep inside and had chicken as their favorite dish.

Standards: HSS.IC.B.6

Click here to practice: Statistics & Probability – Making Inferences & Justifying Conclusions Questions for Grade 10 Math

Domain: Grade 10 >> Statistics & Probability – Conditional Probability & the Rules of Probability

Sample Question: A student council has one upcoming vacancy. The school is holding an election and has eight equally likely candidates. The AP Statistics class wants to simulate the results of the election, so the students have to choose an appropriate simulation method. They intend to do trials with the simulation. Which of these methods would be the most appropriate?

• Spin a wheel with eight equal spaces
• Toss a coin eight times for each election
• Throw a dice
• Throw four die

Answer Explanation: The question states that there are eight equally likely candidates. This means that each candidate has the same chance of winning the election. Only the spinning wheel with eight equal spaces could simulation this situation because the wheel has an equal chance of landing on each space.

Standards: HSS.IC.A.1

Click here to practice: Statistics & Probability – Conditional Probability & the Rules of Probability Questions for Grade 10 Math

Domain: Grade 10 >> Statistics & Probability – Using Probability to Make Decisions

Answer Explanation: Simply count the data points in circles C and E. There are 8 of them out of 24 total data points and by reducing we get 8/24=1/3.

Standards: HSS.CP.B.7

Click here to practice: Statistics & Probability – Using Probability to Make Decisions Questions for Grade 10 Math

Sample Question: A statistician is working for Sweet Shop USA and has been given the task to find out what the probability is that the fudge machine malfunctions messing up a whole batch of fudge in the process. Each malfunction of the machine costs the company $250. The statistician calculates the probability is 1 in 20 batches of fudge will be lost due to machine malfunction. What is the expected value of this loss for one month if the company produces 20 batches of fudge each day?

Answer Explanation: Since most months have 30 days we will assume 30 days in a month. We can use E(x)=x1p1+x2p2+…+xipi or simply calculate as follows E(X)=.05*250*30=$375

Standards: HSS.MD.A.4

Looking for online practice tests? Here is the link to practice more of SBAC Grade 10 Math questions.

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Jenny Watson

2 thoughts on “ grade 10 math questions and solutions ”.

Pingback: Create 10 questions for Grade 1 Algebra with explainations and answer choice - Parent.wiki

https://www.lumoslearning.com/llwp/sample-test-questions/sbac-sample-questions-grade-10-math.html

8/5 = x/576 should read 5/8 = x/576 The rest of the solution is correct so it’s a simple typo.

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Free Printable Math Worksheets for 10th Grade

Math just got more exciting for Grade 10 students! Discover a vast collection of free printable math worksheets by Quizizz, designed to help teachers enhance their students' learning experience and master essential skills.

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Math worksheets for Grade 10 are essential tools for teachers looking to help their students master the challenging concepts in high school mathematics. These worksheets cover a wide range of topics, including algebra, geometry, trigonometry, and probability, ensuring that students have ample practice in each area. Teachers can use these resources to supplement their lesson plans, provide extra practice for struggling students, or even as a basis for group activities in the classroom. By incorporating Grade 10 math worksheets into their curriculum, teachers can ensure that their students are well-prepared for the rigors of high school mathematics and beyond.

Quizizz is an excellent platform for teachers seeking to enhance their Grade 10 math curriculum with engaging and interactive content. In addition to offering a vast collection of Math worksheets for Grade 10, Quizizz also provides teachers with access to a wide variety of quizzes, games, and other educational resources. These materials can be easily integrated into the classroom, allowing teachers to create a dynamic and engaging learning environment for their students. Furthermore, Quizizz offers real-time feedback and analytics, enabling teachers to track student progress and identify areas where additional support may be needed. With Quizizz, teachers have a powerful tool at their disposal to help their Grade 10 students excel in mathematics.

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Grade 10 Math Worksheets: FREE & Printable

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What Do You Learn in 10th Grade Math? Advancing Through Algebra and Geometry

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What Do You Learn in 10th Grade Math? 

Introduction, algebraic mastery, geometry and trigonometry, advanced functions and graphing.

In 10th grade math , students typically expand their knowledge in geometry, exploring more complex theorems and proofs, and diving deeper into algebra with quadratic equations , functions , and inequalities . They may also be introduced to trigonometry, focusing on the relationships between the angles and sides of triangles .

Tenth ( 10th ) grade is a pivotal year in a student’s mathematical journey , where the foundations built in previous years are refined, and advanced concepts are introduced. In this comprehensive guide, we will explore the key concepts and skills that students typically learn in 10th – grade math .

From algebraic mastery to trigonometric functions, we’ll delve into the essential topics and provide practical numerical examples with detailed solutions to illustrate each concept. This in-depth exploration will help students, parents, and educators gain a deeper understanding of the challenging yet rewarding world of 10th – grade mathematics .

Key Concepts and Skills in 10th Grade Math

Tenth graders delve deep into algebra, honing their skills in equations, inequalities, and functions. Key concepts include:

Quadratic Equations

Solving quadratic equations through factoring , completing the square , and using the quadratic formula . For example, solving the equation 2x² – 5x + 3 = 0 .

Systems of Equations

Solving systems of linear equations using various methods such as substitution and elimination . For instance, solving the system:

2x + 3y = 10 4x – y = 5

Polynomial Operations

Performing operations with polynomials , including addition , subtraction , multiplication , and division .

Trigonometric Functions

Introduction to trigonometric functions, including sine, cosine, and tangent. Students learn to apply trigonometry to solve right-triangle problems and explore periodic functions.

Below is an example of a trigonometric function.

Figure-1: Generic Trigonometric Function

Geometry Proofs

Developing geometric proofs , using deductive reasoning to justify statements about geometric figures.

Exponential and Logarithmic Functions

Understanding exponential growth and decay, as well as logarithmic functions . Students learn to solve exponential and logarithmic equations.

Graphing Functions

Mastering the graphing of linear, quadratic, exponential, and trigonometric functions, including transformations and inverse functions. Below is an example of a generic function plotted.

Figure-2: Generic Function

Numerical Examples

Let’s delve into numerical examples to gain a comprehensive understanding of 10th – grade math concepts:

Problem: Solve the quadratic equation 2x² – 5x + 3 = 0.

To solve the quadratic equation, we can use the quadratic formula:

x = (-b ± √(b² – 4ac)) / (2a)

In this case, a = 2, b = -5, and c = 3 .

x = (5 ± √((-5)² – 4 * 2 * 3)) / (2 * 2) x = (5 ± √(25 – 24)) / 4 x = (5 ± √1) / 4

There are two solutions:

x₁ = (5 + 1) / 4 = 6/4 = 3/2 x₂ = (5 – 1) / 4 = 4/4 = 1

So, the solutions are x = 3/2 and x = 1 .

Problem: Find the sine of an angle θ in a right triangle, where the opposite side is 4 unit s and the hypotenuse is 5 units .

Using the definition of sine: sin(θ) = opposite/hypotenuse

sin(θ) = 4/5

Problem: Solve the following system of equations.

We can solve this system using the elimination method. Multiply the second equation by 3 to eliminate y:

Now, add the two equations to eliminate y:

(2x + 3y) + (12x – 3y) = 10 + 15 14x = 25 x = 25/14

Substitute the value of x into the second equation to find y:

4 (25/14) – y = 5 (100/14) – y = 5 – y = 5 – (100/14) – y = (70/14) – (100/14) – y = (-30/14) – y = -15/7

So, the solution is x = 25/14 and y = -15/7.

Exponential Function

Problem: Solve for x in the equation $2^{(x-1)}$ = 8.

Rewrite 8 as a power of 2: 8 = 2³.

Now, we have:

$2^{(x-1)}$ = 2³

Since the bases are the same, the exponents must be equal:

x – 1 = 3 x = 3 + 1 x = 4

So, the solution is x = 4.

Problem: Simplify the expression (3x² – 2x + 5) + (2x² + 4x – 1).

To simplify, combine like terms by adding the coefficients of the same degree terms:

(3x² – 2x + 5) + (2x² + 4x – 1) = (3x² + 2x²) + (-2x + 4x) + (5 – 1) = 5x² + 2x + 4

Problem: Prove that the opposite angles of a parallelogram are congruent.

Let’s consider a parallelogram ABCD where AB || CD and AD || BC.

We want to prove that ∠A ≅ ∠C and ∠B ≅ ∠D.

In ΔABC and ΔADC:

AD = AD (Common side)

AB || CD and AD || BC (Opposite sides of a parallelogram)

Therefore, by the Alternate Interior Angles Theorem, ∠A ≅ ∠C.

In ΔBCD and ΔBAD:

AB = CD (Opposite sides of a parallelogram)

AD || BC and AB || CD (Opposite sides of a parallelogram)

Therefore, by the Alternate Interior Angles Theorem, ∠B ≅ ∠D.

Hence, we have proven that opposite angles of a parallelogram are congruent.

Exponential Growth

Problem: A bacteria colony doubles in size every hour. If there are initially 100 bacteria, how many will there be after 5 hours?

We can use the formula for exponential growth: N(t) = N₀ * 2^(t/h), where N(t) is the final population, N₀ is the initial population, t is the time in hours, and h is the doubling time.

N(5) = 100 * $2^(5/1)$

N(5) = 100 * $2^5$

N(5) = 100 * 32

N(5) = 3200

After 5 hours, there will be 3200 bacteria in the colony.

Trigonometric Identities

Problem: Prove the trigonometric identity: tan²(θ) + 1 = sec²(θ).

We start with the fundamental trigonometric identity:

sec(θ) = 1/cos(θ)

Square both sides:

sec²(θ) = (1/cos(θ))² = 1/cos²(θ)

Now, we use the Pythagorean identity:

1 + tan²(θ) = sec²(θ)

Substitute the value of sec²(θ):

1 + tan²(θ) = 1/cos²(θ)

Rearrange the equation:

1/cos²(θ) – 1 = tan²(θ)

Common denominator:

(1 – cos²(θ))/cos²(θ) = tan²(θ)

Now, use the Pythagorean identity:

sin²(θ) + cos²(θ) = 1

Solve for 1 – cos²(θ):

1 – cos²(θ) = sin²(θ)

Substitute into the equation: sin²(θ)/cos²(θ) = tan²(θ).

tan²(θ) = tan²(θ)

The identity is proven.

Tenth – grade mathematics is a bridge to more advanced mathematical concepts and problem-solving skills. It equips students with a deep understanding of algebra, geometry, trigonometry, and functions, setting the stage for success in higher-level math courses and future academic pursuits.

The numerical examples provided in this guide offer a glimpse into the types of problems and solutions that 10th graders encounter as they navigate the challenging yet intellectually stimulating world of advanced mathematics .

By mastering the intricacies of quadratic equations, systems of equations, trigonometric functions, and exponential functions, students develop critical thinking and analytical skills that extend far beyond the classroom. These skills empower them to tackle real-world problems, make informed decisions, and contribute to fields that rely on mathematical reasoning.

As 10th graders continue to explore and refine their mathematical abilities, they are well-prepared for the exciting mathematical journey that lies ahead in their academic careers.

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Trigonometry Problems and Questions with Solutions - Grade 10

Grade 10 trigonometry problems and questions with answers and solutions are presented.

• The area of a right triangle is 50. One of its angles is 45°. Find the lengths of the sides and hypotenuse of the triangle.
• In a right triangle ABC, tan(A) = 3/4. Find sin(A) and cos(A).
• In a right triangle ABC with angle A equal to 90°, find angle B and C so that sin(B) = cos(B).
• A rectangle has dimensions 10 cm by 5 cm. Determine the measures of the angles at the point where the diagonals intersect.
• The lengths of side AB and side BC of a scalene triangle ABC are 12 cm and 8 cm respectively. The size of angle C is 59°. Find the length of side AC.
• From the top of a 200 meters high building, the angle of depression to the bottom of a second building is 20 degrees. From the same point, the angle of elevation to the top of the second building is 10 degrees. Calculate the height of the second building.
• Karla is riding vertically in a hot air balloon, directly over a point P on the ground. Karla spots a parked car on the ground at an angle of depression of 30°. The balloon rises 50 meters. Now the angle of depression to the car is 35 degrees. How far is the car from point P?
• If the shadow of a building increases by 10 meters when the angle of elevation of the sun rays decreases from 70° to 60°, what is the height of the building?

Solutions to the Above Problems

• x = 10 / tan(51°) = 8.1 (2 significant digits) H = 10 / sin(51°) = 13 (2 significant digits)
• Area = (1/2)(2x)(x) = 400 Solve for x: x = 20 , 2x = 40 Pythagora's theorem: (2x) 2 + (x) 2 = H 2 H = x √(5) = 20 √(5)
• BH perpendicular to AC means that triangles ABH and HBC are right triangles. Hence tan(39°) = 11 / AH or AH = 11 / tan(39°) HC = 19 - AH = 19 - 11 / tan(39°) Pythagora's theorem applied to right triangle HBC: 11 2 + HC 2 = x 2 solve for x and substitute HC: x = √ [ 11 2 + (19 - 11 / tan(39°) ) 2 ] = 12.3 (rounded to 3 significant digits)
• Since angle A is right, both triangles ABC and ABD are right and therefore we can apply Pythagora's theorem. 14 2 = 10 2 + AD 2 , 16 2 = 10 2 + AC 2 Also x = AC - AD = √( 16 2 - 10 2 ) - √( 14 2 - 10 2 ) = 2.69 (rounded to 3 significant digits)
• Use right triangle ABC to write: tan(31°) = 6 / BC , solve: BC = 6 / tan(31°) Use Pythagora's theorem in the right triangle BCD to write: 9 2 + BC 2 = BD 2 Solve above for BD and substitute BC: BD = √ [ 9 + ( 6 / tan(31°) ) 2 ] = 13.4 (rounded to 3 significant digits)
• The triangle is right and the size one of its angles is 45°; the third angle has a size 45° and therefore the triangle is right and isosceles. Let x be the length of one of the sides and H be the length of the hypotenuse. Area = (1/2)x 2 = 50 , solve for x: x = 10 We now use Pythagora to find H: x 2 + x 2 = H 2 Solve for H: H = 10 √(2)
• Let a be the length of the side opposite angle A, b the length of the side adjacent to angle A and h be the length of the hypotenuse. tan(A) = opposite side / adjacent side = a/b = 3/4 We can say that: a = 3k and b = 4k , where k is a coefficient of proportionality. Let us find h. Pythagora's theorem: h 2 = (3k) 2 + (5k) 2 Solve for h: h = 5k sin(A) = a / h = 3k / 5k = 3/5 and cos(A) = 4k / 5k = 4/5
• Let b be the length of the side opposite angle B and c the length of the side opposite angle C and h the length of the hypotenuse. sin(B) = b/h and cos(B) = c/h sin(B) = cos(B) means b/h = c/h which gives c = b The two sides are equal in length means that the triangle is isosceles and angles B and C are equal in size of 45°.
• Let x be the length of side AC. Use the cosine law 12 2 = 8 2 + x 2 - 2 · 8 · x · cos(59°) Solve the quadratic equation for x: x = 14.0 and x = - 5.7 x cannot be negative and therefore the solution is x = 14.0 (rounded to one decimal place).

More References and links on Trigonometry

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Tenth Grade (Grade 10) Problem Solving Strategies Questions

You can create printable tests and worksheets from these Grade 10 Problem Solving Strategies questions! Select one or more questions using the checkboxes above each question. Then click the add selected questions to a test button before moving to another page.

• The cost of her other groceries.
• The number of mangoes she bought.
• The weight of each mango.
• The cost of each mango.
• 525,600 minutes
• 21,900 minutes
• 8,760 minutes
• 913 minutes
• Because this is dimensional analysis, regular rules of math do not apply.
• Since the necessary units cancel out, there is no problem.
• Because each of these factors is equal to one (the numerator and denominator are equal, but in different units).
• By the Multiplicative Property of Equality.
• One variable: C, the cost of moving.
• One variable: t, for the amount of time.
• Two variables: C, for the total cost, and t, for the amount of time.
• Two variables: t, for the amount of time, and d, for the distance traveled.
• The rate of the water's increase, in liters per hour.
• The amount of water in the bucket, in liters.
• The amount of water in the bucket in liters and the amount of time that has passed in hours.
• The size of the bucket in liters and the rate of the drip in liters per hour.
• One variable, the length of the side of the garden.
• One variables, the total perimeter of the garden.
• Two variables, the length and width of the garden.
• Two variables, the amount of fence he has, and the area of the garden.
• Yes, since [math]("N" * "m")/("kg"^2) xx "kg"/1 xx "kg"/1 xx 1/("m") = "N"[/math].
• No, since [math]("N" * "m"^2)/("kg"^2) xx "kg"/1 xx "kg"/1 xx 1/("m") = "N"*"m"[/math].
• No, since [math]("N" * "m"^2)/("kg"^2) xx 1/"kg" xx 1/"kg" xx "m"/1 = ("N" * "m"^3)/ ("kg"^4)[/math].
• No, since [math]("N" * "m"^2)/("kg"^2) xx "kg"/1 xx "kg"/1 xx "m"/1 = "N" * "m"^3[/math].
• Yes, this is the correct model, and there are no more important factors to consider.
• No, this model leaves out many important variables, such as climate, water currents, diseases, and many more.
• No, only the current number of penguins and the historic rate of population growth are important. All the other factors are simply over-complicating the problem.
• Neither correct nor incorrect. Any real world problem can be modeled in multiple ways and with varying degrees of complexity.
• Hockey skill.
• Number of classes a student is taking.
• What type of movies he likes.
• Whether the student intends on attending university after he graduates high school.
• Number of calls placed to 9-1-1 from this neighborhood.
• When calls to 9-1-1 are placed from this neighborhood.
• Names of people arrested in this neighborhood.
• Criminal history of people arrested in this neighborhood.
• Yes, she is correct.
• No. She assumed that all terms need to have the same units, when all terms need to be without units.
• No. Although the variable [math]t[/math] is squared, the units are not. Therefore, the units of the last term are m/s, which are different than the rest of the terms.
• No. Jillian did the dimensional analysis incorrectly. The units of the second term on the right side come out to [math]"m"//"s"^2[/math] and the units of the last term are [math]"m"//"s"^4[/math].
• [math]3 xx 778 xx 26.4 xx 2.5[/math]
• [math]1/3 xx 778 xx 26.4 xx 2.5[/math]
• [math]1/3 xx 778 xx 1/26.4 xx 2.5[/math]
• [math]1/3 xx 778 xx 1/26.4 xx 1/2.5[/math]
• Just one variable, the average cost of a baked good.
• Two variables, one for the cost to make the baked goods, and another for the total revenue of the baked goods.
• Two variables, one for the total number of turnovers sold, and another for the total number of cupcakes sold.
• Two variables, one for the cost of a turnover, and one for the cost of a cupcake.
• Amount of sunlight the plant will get.
• The size of the apartment.
• Whether she will put the plant on the floor or on a table.
• Average temperature of her apartment.
• The average number of tomatoes a tomato plant produces.
• The average weight of tomatoes a tomato plant produces.
• How long it takes to can one pound of tomatoes.
• The average weight of one can of tomatoes.
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10 Strategies for Problem Solving in Math

Created: May 19, 2022

Last updated: January 6, 2024

When faced with problem-solving, children often get stuck. Word puzzles and math questions with an unknown variable, like x, usually confuse them. Therefore, this article discusses math strategies and how your students may use them since instructors often have to lead students through this problem-solving maze.

What Are Problem Solving Strategies in Math?

If you want to fix a problem, you need a solid plan. Math strategies for problem solving are ways of tackling math in a way that guarantees better outcomes. These strategies simplify math for kids so that less time is spent figuring out the problem. Both those new to mathematics and those more knowledgeable about the subject may benefit from these methods.

There are several methods to apply problem-solving procedures in math, and each strategy is different. While none of these methods failsafe, they may help your student become a better problem solver, particularly when paired with practice and examples. The more math problems kids tackle, the more math problem solving skills they acquire, and practice is the key.

Strategies for Problem-solving in Math

Even if a student is not a math wiz, a suitable solution to mathematical problems in math may help them discover answers. There is no one best method for helping students solve arithmetic problems, but the following ten approaches have shown to be very effective.

Understand the Problem

Understanding the nature of math problems is a prerequisite to solving them. They need to specify what kind of issue it is ( fraction problem , word problem, quadratic equation, etc.). Searching for keywords in the math problem, revisiting similar questions, or consulting the internet are all great ways to strengthen their grasp of the material. This step keeps the pupil on track.

Math for Kids

Guess and Check

One of the time-intensive strategies for resolving mathematical problems is the guess and check method. In this approach, students keep guessing until they get the answer right.

After assuming how to solve a math issue, students should reintroduce that assumption to check for correctness. While the approach may appear cumbersome, it is typically successful in revealing patterns in a child’s thought process.

Work It Out

Encourage pupils to record their thinking process as they go through a math problem. Since this technique requires an initial comprehension of the topic, it serves as a self-monitoring method for mathematics students. If they immediately start solving the problem, they risk making mistakes.

Students may keep track of their ideas and fix their math problems as they go along using this method. A youngster may still need you to explain their methods of solving the arithmetic questions on the extra page. This confirmation stage etches the steps they took to solve the problem in their minds.

Work Backwards

In mathematics, a fresh perspective is sometimes the key to a successful solution. Young people need to know that the ability to recreate math problems is valuable in many professional fields, including project management and engineering.

Students may better prepare for difficulties in real-world circumstances by using the “Work Backwards” technique. The end product may be used as a start-off point to identify the underlying issue.

In most cases, a visual representation of a math problem may help youngsters understand it better. Some of the most helpful math tactics for kids include having them play out the issue and picture how to solve it.

One way to visualize a workout is to use a blank piece of paper to draw a picture or make tally marks. Students might also use a marker and a whiteboard to draw as they demonstrate the technique before writing it down.

Find a Pattern

Kids who use pattern recognition techniques can better grasp math concepts and retain formulae. The most remarkable technique for problem solving in mathematics is to help students see patterns in math problems by instructing them how to extract and list relevant details. This method may be used by students when learning shapes and other topics that need repetition.

Students may use this strategy to spot patterns and fill in the blanks. Over time, this strategy will help kids answer math problems quickly.

When faced with a math word problem, it might be helpful to ask, “What are some possible solutions to this issue?” It encourages you to give the problem more thought, develop creative solutions, and prevent you from being stuck in a rut. So, tell the pupils to think about the math problems and not just go with the first solution that comes to mind.

Draw a Picture or Diagram

Drawing a picture of a math problem can help kids understand how to solve it, just like picturing it can help them see it. Shapes or numbers could be used to show the forms to keep things easy. Kids might learn how to use dots or letters to show the parts of a pattern or graph if you teach them.

Charts and graphs can be useful even when math isn’t involved. Kids can draw pictures of the ideas they read about to help them remember them after they’ve learned them. The plan for how to solve the mathematical problem will help kids understand what the problem is and how to solve it.

Trial and Error Method

The trial and error method may be one of the most common problem solving strategies for kids to figure out how to solve problems. But how well this strategy is used will determine how well it works. Students have a hard time figuring out math questions if they don’t have clear formulas or instructions.

They have a better chance of getting the correct answer, though, if they first make a list of possible answers based on rules they already know and then try each one. Don’t be too quick to tell kids they shouldn’t learn by making mistakes.

Review Answers with Peers

It’s fun to work on your math skills with friends by reviewing the answers to math questions together. If different students have different ideas about how to solve the same problem, get them to share their thoughts with the class.

During class time, kids’ ways of working might be compared. Then, students can make their points stronger by fixing these problems.

Check out the Printable Math Worksheets for Your Kids!

There are different ways to solve problems that can affect how fast and well students do on math tests. That’s why they need to learn the best ways to do things. If students follow the steps in this piece, they will have better experiences with solving math questions.

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

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Curriculum  /  Math  /  10th Grade  /  Unit 7: Circles  /  Lesson 10

Lesson 10 of 14

Criteria for Success

Tips for teachers, anchor problems.

• Problem Set

Target Task

Additional practice.

Use angle and side length relationships with chords, tangents, inscribed angles, and circumscribed angles to solve problems. 

Common Core Standards

Core standards.

The core standards covered in this lesson

G.C.A.2 — Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Identify and use angle and segment relationships in circle diagrams to find missing measures. 
• State and describe the conjectures and theorems used to find missing measures in circle diagrams. 

Suggestions for teachers to help them teach this lesson

This lesson summarizes key concepts in the unit; therefore, this lesson should be used as a review day. 

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

In circle $$A$$ below, $$\overleftrightarrow{CE}$$ and $$\overleftrightarrow{DE}$$  are tangent at points  $$C$$ and $$D$$ , respectively.

Find $$m\angle{CED}$$  and  $$m\angle{ADG}$$ .

Guiding Questions

Geometry - 8.10 AP1 by Match Fishtank is made available by GeoGebra  under the CC BY-NC-SA 3.0  license. Copyright © International GeoGebra Institute, 2013. Accessed June 13, 2017, 11:55 a.m..

In the following diagram, the radius of circle  $$D$$  is $$5$$  cm and  $$F$$  is the midpoint of  $$\overline{AE}$$ . The measures of  $$\widehat{EB}$$  and  $$\widehat{BC}$$  are given in the diagram. Find the measures of all other unmarked angles, arcs, and segments.

Module 7: Circles a Geometric Perspective from  Geometry: A Learning Cycle Approach  made available by Mathematics Vision Project  under the CC BY 4.0  license. © 2016 Mathematics Vision Project. Accessed Oct. 19, 2017, 2:58 p.m..

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

In the diagram below,  $$\overline{PA}$$ is tangent to circle $$O$$ , and  $$\overline{AB}$$ is a chord. If $$m\widehat{ACB}=300$$ , find the measure of $$\angle BAP$$ .

G.C.A.2: Chords, Secants, and Tangents 17 is made available on JMAP by Steve Sibol and Steve Watson. Copyright © 2017  JMAP, Inc. - All rights reserved. Accessed Sept. 19, 2018, 10:43 a.m..

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• EngageNY Mathematics Geometry > Module 5 > Topic C > Lesson 16
• Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 5, "From Polygons to Circles"
• Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 6, "Circular Reasoning"
• MARS Formative Assessment Lessons for High School Solving Problems with Circles and Triangles
• Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 3, "Cyclic Polygons"

Topic A: Equations of Circles

Derive the equation of a circle using the Pythagorean Theorem where the center of the circle is at the origin.

G.GPE.A.1 G.GPE.B.4

Given a circle with a center translated from the origin, write the equation of the circle and describe its features. 

G.C.A.1 G.CO.A.5 G.GPE.A.1

Write an equation for a circle in standard form by completing the square. Describe the transformations of a circle. 

G.CO.A.5 G.GPE.A.1

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Topic B: Angle and Segment Relationships in Inscribed and Circumscribed Figures

Define a chord to derive the Chord Central Angles Conjecture and Thales’ Theorem. 

Describe the relationship between inscribed and central angles in terms of their intercepted arc. 

Determine the angle and length relationships between intersecting chords. 

Prove properties of angles in a quadrilateral inscribed in a circle.

Define and determine properties of tangents and secants of circles to solve problems with inscribed and circumscribed triangles. 

G.C.A.2 G.C.A.3

Construct tangent lines to a circle to define and describe the circumscribed angle. 

G.C.A.2 G.C.A.4

Topic C: Arc Length, Radians, and Sector Area

Define, describe, and calculate arc length.

Describe the proportional relationship between arc length and the radius of a circle. Convert between degrees and radians to write the arc measure in radians. 

Calculate the sector area of a circle. Identify relationships between sector area, arc angle, and radius. 

Use sector area of circles to calculate the composite area of figures.

G.C.B.5 N.Q.A.2 N.Q.A.3

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