A possible set of scores for the students is: 4, 4, 6, 6, 6
How to solve the measures of Central Tendencies?We are given that five students took the quiz. If the lowest score is 4 and the highest score is 6, we can say that the set of values can be expressed as:
4, x, y, z, 6
Here we can see that the mode is 6 and the mean is 5.2. Therefore, at least one other student of hers has scored her 6, which could result in:
4, x, y, 6, 6
Therefore:
(4 + x + y + 6 + 6)/5 = 5.2
16+x+y=26
x + y = 10
So the other two numbers are probably 6 and 4, because 6 is the mode, so they can't both be 5.
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If the ratio of shirts to jeans is 5 : 3 and the no. of shirts are 15, what is the no. of jeans?
Answer:
9
Step-by-step explanation:
let x be number of jeans
3/5 = x/15
x=(3/5)15
x=9
Answer:
9 jeans
Step-by-step explanation:
The 5 part of the ratio relates to 15 shirts , then
15 ÷ 5 = 3 ← value of 1 part of the ratio , then
3 parts = 3 × 3 = 9 ← number of jeans
Mrs. Smedema likes cookies. Each student in her 8th grade math class presented her with a cookie. One-third of the students each gave her an oatmeal cookie, one-sixth of the students each gave her a cinnamon cookie, one-fourth of the students each gave her a chocolate chip cookie, and the remaining 9 students each gave her a peanut butter cookie. How many students are in Mrs Smedema’s 8th grade math class?
Late answer, but whatever.
Answer:
36. There are 36 students in the class.
Step-by-step explanation:
4/12 oatmeal
2/12 cinnamon
3/12 chocolate
9 peanut
The fractions added together = 9/12
This means the 9 students = 3/12 of the class
3 x 4 = 12
9 x 4 = 36
How many 6-digit numbers are there in all?
Answer:
900,000 6
∴ there are 900,000 6-digit numbers in all.
What’s the answer???
Answer:
10 stickers on one sheet
Step-by-step explanation:
96 - 16 ÷ 8 =
80 ÷ 8 = 10
have a nice day! :)
Boats from all along the Atlantic coast dock at a busy marina. Of the first 10 boats to dock at the marina one day, 5 were from Connecticut. Considering this data, how many of the next 12 boats to dock should you expect to be from Connecticut?
Answer:
8
Step-by-step explanation:
The expected number of boats of the next 12 boats to dock is from Connecticut is 6.
What is Probability?Probability is simply the possibility of getting an event. Or in other words, we are predicting the chance of getting an event.
The value of probability will be always in the range from 0 to 1.
Given that,
Boats from all along the Atlantic coast dock at a busy marina.
Of the first 10 boats to dock at the marina one day, 5 were from Connecticut.
Probability that the boats which dock are from Connecticut = 5/10 = 1/2
When the next 12 boats also arrive,
Expected number of boats from Connecticut = 1/2 × 12 = 6
Out of 12, 6 boats are from Connecticut.
Hence the expected number of boats is 6.
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please help i have a test tomorrow
Answer:
a
b
e
Step-by-step explanation:
remember length times width equals area
Carmen has 8 popsicles in her freezer: 3 grape, 2 cherry, 2 orange and 1
strawberry. If Carmen takes one popsicle out without looking, eats it, and then
takes another popsicle out without looking, what is the probability that both
popsicles will be orange?
Answer:
2/8 or 1/4
Step-by-step explanation:
241.78 divided by (-3.85)
Answer:
-62.8 is the Answer fot the equation, Hope this helped
Consider a random sample from a normally distributed population of large size. i. If the population variance o2 = 35, what sample size is needed to estimate the mean within +2 with 99% confidence? ii. If instead we would like to estimate some true proportion, what sample size is needed to estimate the true proportion within 22% with 99% confidence? Now consider a random sample from a population of large size with unknown distribution. iii. If the population variance o2 50, what sample size is needed to estimate the mean within +1 with 95% confidence (using the 22.5% value)? iv. Why is it the case that such estimating process is still legitimate?
i. a sample size of 138 is needed to estimate the mean within +2 with 99% confidence. Sample size needed: 138
ii. Sample size needed: 342
iii. Sample size needed: 193
iv. Estimating process is legitimate due to the Central Limit Theorem.
What is normal distribution?
Normal distribution, also known as the Gaussian distribution or bell curve, is a continuous probability distribution that is symmetric and characterized by its mean and standard deviation.
i. To estimate the mean within +2 with 99% confidence, we can use the formula for the sample size needed for estimating the population mean:
[tex]n = (Z * \sigma / E)^2[/tex]
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (99% confidence corresponds to a Z-score of approximately 2.576)
σ = population standard deviation (given as √35 since [tex]o^2[/tex] = 35)
E = maximum error tolerance (+2 in this case)
Substituting the values into the formula:
[tex]n = (2.576 * \sqrt{35} / 2)^2 = 137.13[/tex] (approx)
Since the sample size needs to be a whole number, we round up to the nearest integer. Therefore, a sample size of 138 is needed to estimate the mean within +2 with 99% confidence.
ii. To estimate the true proportion within 22% with 99% confidence, we can use the formula for the sample size needed for estimating the population proportion:
[tex]n = (Z^2 * p * (1 - p)) / E^2[/tex]
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (99% confidence corresponds to a Z-score of approximately 2.576)
p = estimated proportion (0.5 is commonly used for unknown proportions)
E = maximum error tolerance (22% in this case, which is 0.22)
Substituting the values into the formula:
[tex]n = (2.576^2 * 0.5 * (1 - 0.5)) / 0.22^2 = 341.28[/tex]
Since the sample size needs to be a whole number, we round up to the nearest integer. Therefore, a sample size of 342 is needed to estimate the true proportion within 22% with 99% confidence.
iii. When the population variance [tex]o^2[/tex] is unknown, we can use the t-distribution instead of the Z-distribution for estimating the mean. The formula for the sample size needed for estimating the population mean with an unknown variance is:
[tex]n = (t * \sigma / E)^2[/tex]
Where:
n = sample size
t = t-score corresponding to the desired confidence level and degrees of freedom (in this case, for 95% confidence and a large sample size, t can be approximated as 1.96)
σ = estimated standard deviation (given as √50 since [tex]o^2[/tex] = 50)
E = maximum error tolerance (+1 in this case)
Substituting the values into the formula:
[tex]n = (1.96 * √50 / 1)^2 = 192.08[/tex]
Since the sample size needs to be a whole number, we round up to the nearest integer. Therefore, a sample size of 193 is needed to estimate the mean within +1 with 95% confidence using the 22.5% value.
iv. The estimating process is still legitimate in this case because the sample size is large and the Central Limit Theorem applies. The Central Limit Theorem states that for a large enough sample size, the sampling distribution of the mean (or proportion) will be approximately normally distributed, regardless of the shape of the population distribution. This allows us to make inferences about the population mean or proportion using sample statistics. Additionally, the use of the t-distribution accounts for the uncertainty introduced by using the sample standard deviation instead of the population standard deviation.
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if y=4x^2 −3 , what is the minimum value of the product \displaystyle xyxy ?
The minimum value of the product xyxy is -9, obtained when y equals -3.
To find the minimum value of the product xyxy, we need to first determine the minimum value of the expression y = 4x² - 3.
The given expression is a quadratic equation in the form of y = ax² + bx + c, where a = 4, b = 0, and c = -3.
To find the minimum value, we need to determine the x-coordinate of the vertex of the parabola, which corresponds to the minimum point.
The x-coordinate of the vertex can be found using the formula: x = -b / (2a).
In this case, b = 0 and a = 4, so the x-coordinate of the vertex is x = -0 / (2 × 4) = 0.
Substituting the x-coordinate back into the equation y = 4x² - 3, we can find the minimum value of y.
y = 4(0)² - 3 = -3.
Therefore, the minimum value of y is -3.
To find the minimum value of the product xyxy, we can substitute the minimum value of y (-3) back into the expression:
xyxy = x × (-3) × x × (-3) = -9x².
Since the coefficient is negative, the minimum value of the product xyxy is -9.
Hence, the minimum value of the product xyxy is -9.
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Please help ASAP!
Find the area of the figure.
Number 5.
Answer:
70 ft squared
Step-by-step explanation:
Separate into rectangle and triangle then add the found areas together for total area.
rectangle...
10×4=40
triangle...
[tex] \frac{bh}{2} = [/tex]
(12×5)/2=
60/2=30
add area for rectangle and triangle...
40+30=70
Pls help, question on picture, will do brainliest if right
no links!!!!!
Answer:
[tex]\boxed {\boxed {\sf sin(\theta)=\frac {16}{20}}}[/tex]
Step-by-step explanation:
We are asked to find the sine for the angle indicated. Remember that sine is equal to the opposite over the hypotenuse.
sin(θ)= opposite/hypotenuseAnalyze the triangle given. We have the 2 legs (16 and 12), but we do not have the hypotenuse (the longest side). We must solve for it.
Since this is a right triangle, we can use the Pythagorean Theorem.
[tex]a^2+b^2=c^2[/tex]
Where a and b are the legs and c is the hypotenuse. We know 16 and 12 are the legs.
[tex](16)^2+(12)^2=c^2[/tex]
Solve the exponents.
16²= 16*16=25612²= 12*12= 144[tex]256+144=c^2[/tex]
[tex]400=c^2[/tex]
Since we are solving for c, we must isolate the variable. It is being squared, so we take the inverse: a square root.
[tex]\sqrt{400}=\sqrt{c^2} \\20=c[/tex]
Now we know the hypotenuse is 20. The side opposite of the angle θ is 16.
opposite= 16 hypotenuse=20[tex]sin (\theta)= \frac {opposite}{hypotenuse} \\sin (\theta)= \frac{16}{20}[/tex]
The sine of the angle is equal to 16/20. This can be reduced to 4/5 if necessary (by dividing the numerator and denominator by 4).
Can someone help me pls
Answer:
The answer would be C
Step-by-step explanation:
This is because 3/4 times two is 6/4 simplified is 1 1/2
Mr. Hart asked students to find the volume of the triangular prism below. Which student wrote the correct expression for the value of B, the area of the base?
Answer:
Omar
Step-by-step explanation:
Given
See attachment for prism
Required
Whose expression is correct
The base of the attached prism is a triangle that has the following dimension.
[tex]Base =7cm[/tex]
[tex]Height = 24cm[/tex]
So, the area is:
[tex]Area = \frac{1}{2} * Base * Height[/tex]
[tex]Area = \frac{1}{2} * 7 * 24[/tex]
Comparing the expressions of Omar and Kari, we can conclude that Omar is correct
Find the volume. I hate these questions!!
please give me brainiest i was working on it the whole time.
Answer: 4* 2 3/4* 8 1/2= 93 1/2
Find the total surface area of the rectangular prism
Answer:
124
Step-by-step explanation:
To find the surface area, we add the area from all faces or the prism.
the first face, we can put 8 times 2.5, which is 20. There are two of these faces, so we have 40 in our area so far.
next the other face. 4 times 2.5 is 10, and there are two of these faces. 60 in out area.
now the top and bottom faces. 8 times 4 is 32. there are two of these faces, and 60 +64 = 124
Scientists were monitoring the temperature of a solution. It began at 63°F, and the temperature changed by 8°F over the course of 6 hours.
Use this information to complete this statement.
The final temperature of the solution was a minimum of
°F and a maximum of
°F.
Scientists were monitoring the temperature of a solution. It began at 63°F, and the temperature changed by 8°F over the course of 6 hours.
Use this information to complete this statement.
The final temperature of the solution was a minimum of
°F and a maximum of
°F.
Answer:
minimum is 55
maximum is 71
Step-by-step explanation:
got this from another answer so i hope it helps you
Mr. Alberto Blanco is a 70 years-old widower. His wife of 44 years died two years ago. He has three sons aged 34, 38 and 41 and a daughter aged 43. He is a retired electrician. He lives alone in his own apartment. His relationship with his children is marked by a considerable degree of conflict. Mr. Blanco is currently hospitalized with gangrene of his right foot and lower leg. Problems with his foot began three years ago when he had an infection in one toe that became gangrenous. At that time it was discovered that he was diabetic. The toe was amputated. Last year he bruised his right foot while getting into a bus. The bruise developed into gangrene that resulted in an operation 6 months ago in which a portion of his right foot was amputated. At that time, an arterial bypass was done to decrease the likelihood that gangrene would recur. He went to a rehabilitation center where he remained for five months. Last week it was discovered that he had gangrene in the remainder of the right foot and part of the lower leg. He was readmitted to the hospital. He originally agreed to amputation of the leg but on the same morning the surgery was scheduled he verbally withdrew the consent. He discussed with some people the reason for his decision. He does not want to be a burden to his children. Mr. Blanco is discouraged by the failure of the earlier operations. He does not believe the operation will cure him and does not want to live as a dependent person or in a nursing home. Mr. Blanco assures that he does not fear death. Although quiet and stoic, he tends to be stubborn and somewhat irascible when pressured. Guide questions for case: Alberto Blanco 1. What are the ethical principles pertinent to this case? 2. Should Mr. Blanco have the right to refuse treatment even if the consequence may be that he will die? 3. Could he be seriously depressed and not mentally competent to make that decision? 4. May the doctor go ahead with the surgery, given the fact that Mr. Blanco had previously given his written consent for the procedure? 5. Would your decision be different if Mr. Blanco were refusing treatment for another condition such as terminal cancer or irreversible heart-failure?
Mr. Blanco has the right to refuse treatment, even if it means he will die. The doctor should respect Mr. Blanco's decision and engage in open and honest communication with him to understand his concerns.
1. The ethical principles pertinent to this case include autonomy, beneficence, non-maleficence, and respect for patient's wishes. Autonomy refers to Mr. Blanco's right to make decisions about his own healthcare. Beneficence and non-maleficence require the healthcare professionals to act in Mr. Blanco's best interest and do no harm. Respect for patient's wishes entails honoring his decision and preferences.
2. Yes, Mr. Blanco should have the right to refuse treatment even if the consequence may be that he will die. It is his autonomy and right to make decisions about his own body and medical treatment. As long as he is mentally competent, he has the right to make choices about his healthcare, even if they go against medical advice.
3. It is important to consider the possibility that Mr. Blanco may be seriously depressed and not mentally competent to make the decision. In such cases, a thorough evaluation of his mental capacity and emotional state should be conducted by healthcare professionals to ensure that he is able to make an informed decision. If he is found to lack mental competence, appropriate steps should be taken to protect his best interests.
4. Given that Mr. Blanco had previously given his written consent for the surgery but verbally withdrew it later, the doctor should respect his current decision. A patient's consent can be withdrawn at any time, and healthcare professionals should prioritize the patient's current wishes. It is essential to engage in open and honest communication with Mr. Blanco to understand his concerns and explore alternative options if available.
5. The decision may be different if Mr. Blanco were refusing treatment for another condition such as terminal cancer or irreversible heart failure. In such cases, the prognosis and potential outcomes may be different, and the balance between preserving life and respecting autonomy may shift. Each case should be evaluated individually, considering the patient's medical condition, prognosis, quality of life, and their values and preferences. Open discussions with the patient, their family, and the healthcare team are crucial in reaching the most ethically appropriate decision.
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Given that the point P(6.-7) lies on the line 6x + ky = -20, find k Need Help? Let p be the function defined by g(x) = -x2 + Bx. Find g(a + h), g(-a), g(sqrt a) a + g(a), and 1/g(a)
The required solutions are:
[tex]g(a + h) = -(a + h)^2 + B(a + h)[/tex]
[tex]g(-a) = -(-a)^2 + B(-a)[/tex]
[tex]g(\sqrt{(a)}) = -(\sqrt{(a)})^2 + B(\sqrt{(a)})[/tex]
[tex]a + g(a) = a + (-a^2 + Ba)\\[/tex]
[tex]1/g(a) = 1/(-a^2 + Ba)[/tex]
To find the value of k, we can substitute the coordinates of the point P(6, -7) into the equation of the line and solve for k.
Given: P(6, -7) and the line equation 6x + ky = -20
Substituting the x and y values of P into the equation, we have:
6(6) + k(-7) = -20
36 - 7k = -20
Now, let's solve for k:
-7k = -20 - 36
-7k = -56
k = (-56)/(-7)
k = 8
Therefore, the value of k is 8.
To solve the second part of your question, let's work with the function [tex]g(x) = -x^2 + Bx.[/tex]
1. g(a + h):
Substitute (a + h) into the function:
[tex]g(a + h) = -(a + h)^2 + B(a + h)[/tex]
Simplify the expression as needed.
2. g(-a):
Substitute (-a) into the function:
[tex]g(-a) = -(-a)^2 + B(-a)[/tex]
Simplify the expression as needed.
3. [tex]g(\sqrt{(a)})[/tex]:
Substitute [tex]\sqrt{(a)}[/tex] into the function:
[tex]g(\sqrt{(a)}) = -(\sqrt{(a)})^2 + B(\sqrt{(a)})[/tex]
Simplify the expression as needed.
4. a + g(a):
Substitute 'a' into the function and add it to a:
[tex]a + g(a) = a + (-a^2 + Ba)[/tex]
Simplify the expression as needed.
5. 1/g(a):
Take the reciprocal of g(a):
[tex]1/g(a) = 1/(-a^2 + Ba)[/tex]
Simplify the expression as needed.
Please note that without the specific value of B or the variable a, we can only provide general expressions for each of the given calculations.
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Convert 73.5% to a fraction in lowest terms.
Answer:
147/200
Step-by-step explanation:
we compute the deviation of the second observation in the data set from the mean, and find that the result is a negative number. this tells us that
When we compute the deviation of the second observation in the data set from the mean, and find that the result is a negative number, this tells us that there is good reason to use the median as opposed to the mean as a measure of central tendency.
Computing the deviationIn computing deviations, it is important to note that extremely skewed deviations can increase the normal distributions and make it difficult to use the standard deviation as a measure of central tendency.
So, when the deviation of the second observation is a negative number, the distribution will be affected and it may become better to use the median as a measure of central tendency.
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Complete Question:
We compute the deviation of the second observation in the data set from the mean, and find that the result is a positive number. This tells us that:
there is a positive overall skewness in the data set.
there is good reason to use the median as opposed to the mean as a measure of central tendency.
the first deviation must also be positive.
the second observation is greater than the sample average.
I manage to run a mile in 6 minutes. What is my average speed in miles per hour.
Answer:
10 miles per hour
Step-by-step explanation:
If you run a mile in 6 minutes...
10miles x 6minutes is 60 minutes. An hour.
You can run 10 miles in the hour at that rate. Does that make sense?
assume that in blackjack, an ace is always worth 11, all face cards (jack, queen, king) are worth 10, and all number cards are worth the number they show. given a shuffled deck of cards:
a) Probability of drawing two cards summing to 21: 1/331
b) Probability of drawing two cards summing to 10: 28/1326
c) Probability of drawing a third card to make the sum strictly larger than 21: 2/25
a) To calculate the probability of drawing two cards that sum to 21, we need to determine the number of favorable outcomes and the total number of possible outcomes.
There are four ways to get a sum of 21 with two cards: drawing an Ace and a face card (4 possibilities).
The total number of possible outcomes is given by the combination of choosing two cards from a deck of 52 cards, which is C(52, 2) = 1326.
Therefore, the probability of drawing two cards that sum to 21 is 4/1326, which simplifies to 1/331 or approximately 0.0030.
b) To calculate the probability of drawing two cards that sum to 10, we need to determine the number of favorable outcomes.
There are several combinations that sum to 10: drawing a 4 and a 6, drawing a 5 and a 5, drawing a 6 and a 4, and drawing a face card and a 10.
The total number of possible outcomes remains the same, which is 1326.
Therefore, the probability of drawing two cards that sum to 10 is (4 + 4 + 4 + 16)/1326, which simplifies to 28/1326 or approximately 0.0211.
c) Given that you have already drawn the 10 of clubs and the 4 of hearts, the sum of these two cards is 10 + 4 = 14.
To find the probability that the sum of all three cards is strictly larger than 21, we need to consider the remaining cards in the deck. Since there are 50 cards left in the deck, we calculate the probability of drawing a card that makes the sum exceed 21.
The only way to exceed 21 is to draw an Ace, which is worth 11. There are four Aces in the deck.
Therefore, the probability of drawing a card that makes the sum strictly larger than 21 is 4/50, which simplifies to 2/25 or approximately 0.08.
Complete question:
Assume that in blackjack, an ace is always worth 11, all face cards (jack, queen, king) are worth 10, and all number cards are worth the number they show. given a shuffled deck of cards:
a) What is the probability that you draw two cards and they sum 21
b) What is the probability that you draw two cards and they sum 10
c) Suppose, you have drawn two cards: 10 of clubs and 4 of hearts. You now draw a third card from remaining 50. What is the probability that the sum of all three cards is strictly larger than 21?
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What is the result of subtracting the second equation from the first ? -7x-y=0 and 7x+8y=-6
Answer:
- 14x - 9y = 6
Step-by-step explanation:
- 7x - y = 0 → (1)
7x + 8y = - 6 → (2)
Subtracting (2) from (1) term by term
- 7x - 7x - y - 8y = 0 - (- 6)
- 14x - 9y = 0 + 6
- 14x - 9y = 6
*
4. HINT: Remember the "square" means 90 degrees
25 points
The circle with center O has a circumference of 36 units. What is
the length of minor arc AC?
A. 9 units
B. 12 units
C. 18 units
D. 36 units
joshua is planting a triangular garden he sketched his diagram measurements are 12m and 10m the questions asks to find the missing side PLZ THIS IS URGENT
Answer:
It's 16m
Step-by-step explanation:
Let's say the missing side is C, and since the triangle is right-angle triangle, you can use Pythagoras theorem. Which is
C squared is equal to A squared plus B squared
Like this: c^2 = a^2+b^2
Then
c^2= (10)^2 + (12)^2
c^2= 100 + 144
c^2= 244
c=
[tex] \sqrt{244} [/tex]
c = 15.6
Hope this helps :)
Determine whether the data described below are qualitative or quantitative and explain why. The preferred hands of an experiment's participants.
A. The data are quantitative because they consist of counts or measurements.
B. The data are qualitative because they don't measure or count anything.
C. The data are quantitative because they don't measure or count anything.
D. The data are qualitative because they consist of counts or measurements.
The correct answer is D. The data are qualitative because they consist of counts or measurements.
Qualitative data refers to non-numerical information that is categorical or descriptive in nature. It describes qualities or characteristics of a subject.
In this case, the preferred hands of the experiment's participants would be categorical information, as it describes the quality or characteristic of which hand the participants prefer to use.
On the other hand, quantitative data refers to numerical information that can be measured or counted. It involves quantities or amounts.
Examples of quantitative data would include measurements such as height, weight, or number of participants.
Since the preferred hands in this scenario are categorical and not measured or counted, the data is qualitative. It focuses on the qualities or categories of left-handedness or right-handedness, rather than measuring or counting any specific quantities.
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11. Michelle used place value to multiply 4,126 and 87. Which is not a step that Michelle could have
taken in her work?
A Multiply 4,126 x 7
B. Multiply 4,126x8
C. Multiply 4,126 x 8 tens
D. Add 28.882 330,080
Answer:
b
Step-by-step explanation:
You have 64 grams of a radioactive kind of strontium. If its half-life is 6 hours, how much will be left after 3 hours?
Answer: 48 grams
Step-by-step explanation:
The half-life of an element refers to the amount of time that it takes for half of it to decay.
For instance, if the half life is 2 hours and the element weighs 12 g, the element will only weigh 6g after 2 hours.
The half-life of the above is 6 hours, in 3 hours therefore, it would only have lost of the half that it would have lost in 3 hours which means it would have lost only a quarter.
= 64 * ( 1 - 1/4)
= 48 grams
Suppose the learning activity by Teacher Thato and all the three assessment of learning activities by the student teachers captures the entire learning experience of Teacher Thato's learners. Discuss the extent to which Teacher Thato's learners were exposed to opportunities for achieving mathematical proficiency (Kilpatrick, Swafford & Findell, 2001).
If the activities and assessments were designed to facilitate learning and encourage higher-order thinking skills, learners were exposed to opportunities for achieving mathematical proficiency (Kilpatrick, Swafford & Findell, 2001).
The teaching and learning process revolves around assessing the level of knowledge students have and guiding them to achieve specific goals.
The role of assessments in the classroom is to measure students' understanding of the taught content and to provide feedback to the teacher on areas where improvement is required.
Suppose the learning activity by Teacher Thato and all the three assessment of learning activities by the student teachers captures the entire learning experience of Teacher Thato's learners.
The extent to which Teacher Thato's learners were exposed to opportunities for achieving mathematical proficiency depends on the effectiveness of the teaching and learning process.
Mathematical proficiency is the knowledge and skills that learners gain through regular interactions with mathematical concepts.
Mathematical proficiency means understanding mathematics as well as being able to apply mathematical knowledge in real-world scenarios.
Therefore, if Teacher Thato's learning activities and assessment methods were designed to provide learners with opportunities to apply mathematical concepts to real-world problems, then they were exposed to opportunities for achieving mathematical proficiency (Kilpatrick, Swafford & Findell, 2001).
Similarly, if the activities and assessments required learners to engage in critical thinking, problem-solving, and other higher-order thinking skills, then it's possible that learners were exposed to opportunities for achieving mathematical proficiency (Kilpatrick, Swafford & Findell, 2001).
In summary, if the activities and assessments were designed to facilitate learning and encourage higher-order thinking skills, learners were exposed to opportunities for achieving mathematical proficiency (Kilpatrick, Swafford & Findell, 2001).
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