• problems may be written in words or using numbers and variables.
• problem solving includes examining the question to find the key ideas,
choosing an appropriate strategy, doing the maths,
finding the answer and then re-checking.
EXAMPLES:
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A question that needs a solution. In mathematics some problems use words: "John was traveling at 20 km per hour for half an hour. How far did he travel?" And some use equations: "Solve x+5=22"
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Addition and subtraction, multiplication and division, equation-related mathematical problems, problem|definition & meaning.
A mathematical problem is an unsolved question . These problems usually provide some values and ask to find some unknown value . For example, if you cycled a total distance of 16 km in one hour, then what was your average speed? In other problems, you might have an equation already and need to solve it for the unknown variable e.g., what is the value of x in x + 6 = 21?
Some mathematical Problems include the usage of words , such as, “Jim maintained a speed of 30 kilometers per hour during the entire hour . How much ground did he cover?” Others use equations such as “If x + 19 = 78, what is x?” In this article, we shall talk about them in great detail.
Figure 1 – Problems in Mathematics
Word problems are notoriously difficult for pupils to master. Because there are so many moving parts in the process of solving word problems , it can be difficult to isolate the specific factor that is making things difficult for students.
Figure 2 – Types of Word Problems
There are three primary categories of problems involving addition and subtraction:
Any problem in which you begin with one quantity , acquire some more, and then finish with a greater quantity is said to be a joining problem. Any situation in which you begin with one quantity , remove some of that quantity , and then finish up with a smaller quantity is known as a separating problem. Take, for instance:
In each of these scenarios, the end outcome is unknown because we know how much we begin with, we know how much is joined or separated (the change), and now we need to determine how much is left over after the process . Both of these issues may be solved using the following straightforward pattern:
Students need to solve issues in which either the result, the change, or the starting point is unknown .
You’ll find that there are innumerable opportunities throughout the day to utilize the terminology of joining and separating! The most obvious of them is eating because your child is always considering joining other children in order to grab more of their food and then consuming that food after it has been obtained (separating).
When you play a game with your child that involves blocks, ask them to count how many are in their tower. Then you should ask them, “ Now I’ve added three more blocks to your tower . How many blocks do you currently have on your tower?” Have your child count the number of cars in a row while they see vehicles parked in a parking lot. Then pose the following question: “If two vehicles depart the parking lot, how many vehicles will be left?”
If your child is having trouble, you shouldn’t give them the solution. Instead, you should assist them in carrying out the action.
“You have 9 blueberries. Now you are going to devour four of them. When you consume those blueberries, what happens to the blueberries? Your kid might remark something along the lines of “They went in my stomach!” And you are able to reply, “Good.” So, those blueberries are still sitting there on your plate, are they?”
They will respond with a negative, at which point you can say, “Ok. So please display those blueberries that are disappearing from your dish. When performing an additive comparison, the problem might have the following forms, where x could be any whole number.
There are three primary kinds of word problems involving multiplication and division
When there is the same number of items in two different groups, we refer to such groupings as “equal groups.” Therefore, an equal number of items or things are grouped together in each equal group.
For example: If there are three boxes and you put five candies in each box, then there will be an equal number of candies in each box. At this point, we will assume that there ar e three equal groupings , each containing five candies .
The operations of multiplication and division can be represented by using rows and columns in an array. The rows denote the number of different categories. The number of items in each category, as well as their respective sizes, are denoted by the columns, although this is not necessarily a strict rule, and the two can be swapped.
It is essential for one to keep in mind that rows, which represent groups, are drawn horizontally, and columns, which represent the number of items in each group, are drawn vertically.
When doing a multiplicative comparison, the problem can involve expressions such as where x can be any whole number.
It is possible that the product, the size of the group, or the number of groups will all be unknown within each type of problem. Once again, we make use of counters to guide the students through the process of selecting the appropriate equation to apply while trying to solve the various kinds of issues.
The difference between issues involving addition and subtraction and problems involving multiplication and division should be brought to the attention of the students.
Students’ thinking and ability to solve problems are considerably improved when they are required to compose word problems in a fashion that is consistent with a certain word problem style. Because this is a considerably more difficult ability, the first time we practice composing word problems, we usually do so under the supervision of an instructor.
Equations are used to solve issues , and in order to solve a problem using equations, we must do two things:
One of the most notable characteristics of an algebraic solution is that the quantity that is being sought is incorporated into the very operation that is being performed. Because of this, we are able to construct a statement of the conditions in the same form as if the problem had already been solved.
Figure 3 – Equation Related Mathematical Problems
Nothing else has to be done at this point other than to simplify the equation and determine the total sum of the quantities that are already known. Because they are equivalent to the unknown quantity on the opposite side of the equation , the value of that is likewise determined, which means that the problem has been solved as a result.
These examples are quite literally examples of problems and illustrate the process of translating words into equations and then simplifying them to find the value of the unknown variable.
When a given integer is divided by 10, the sum of the quotient, dividend, and divisor equals 54. Determine the number that satisfies it.
Let x equal the desired number. Then:
(x / 10) + x + 10 = 54
x + 10x + 100 = 540
11x = 540 – 100
When a deal is made, a particular amount of profit or loss is realized by a merchant. In the second deal, he makes a profit of 250 dollars, but in the third, he loses 50 dollars. In the end, he determines that the three transactions resulted in a profit of one hundred dollars for him. In comparison to the first, how much ground did he make or lose?
In this particular illustration, the profit and the loss are of opposing characters, so it is necessary to differentiate between them using signs that are the opposite of one another. When the profit is a plus sign (+), the loss should be a minus sign (-).
Let’s say x equals the total amount needed.
The conclusion that follows from this is that x plus 250 minus 50 equals 100.
So, x = -100 .
The fact that the answer has a negative sign attached to it demonstrates that there was a loss incurred in the initial transaction; hence, the correct sign for x is also a negative sign. However, because this is dependent on the response, leaving it out of the calculation won’t result in an error at all.
All images were created with GeoGebra.
Parents and several dozen young children have filtered into the HOLA Ohio Hispanic Community Center in Painesville on a Wednesday night in late February as a small crew of volunteers gets busy.
Some of the volunteers pull out worksheets and pencils, while others place pitchers of vividly hued hibiscus tea on tables and are busy preparing food. On the menu: quesadillas with steak or chorizo, rice, beans and homemade salsas.
The HOLA Ohio Community Center opened in 2022. It was long a dream of Executive Director Veronica Isabel Dahlberg, who herself is a child of immigrants from Hungary and Mexico. She’s watched as families have immigrated to Painesville from Mexico for work at the nearby tree nurseries for three decades, swelling the population of English learners at the Painesville City School District.
"We were just meeting under a canopy in the park and didn't really have a space," Dahlberg said. "But I was watching the state of Ohio school report card for the Painesville schools and the very dismal rankings. And it really worried me a lot. And so that was one of the driving factors behind establishing the community center."
The school district has the highest percentage of Hispanic students and English language learners in all of Ohio, about 57% and 27% of the student population respectively, according to the district, and many of those students are struggling academically. The district approached HOLA Ohio last year, asking Dahlberg if they could partner to create regular math tutoring nights during the school year, said Wendy Camper, director of curriculum at Painesville schools.
"Actually, we're lower performing in math than we are in literacy, which sometimes, it's hard to believe as a country," she explained. "But it's true. And statistically, children drop out of high school not because of English language arts, science or social studies: it's because of math."
After dinner, it’s time for students and their tutors to get to work. Tables are organized by grade level, where volunteers like Monica Garza are helping students with basics, like counting using dinosaurs and fake coins. Garza, president of the Spanish club at Mentor High School, is working with first grader Nathaly.
"She likes to count coins," Garza said, laughing.
The tutoring nights rely on volunteers like Garza.
"It's just really sweet that it's basically trusting in yourself that you can help other people, and that these people will benefit from your help," Garza said.
Nathaly’s parents Brenda Cordova and Alvaro Medina have been in the U.S. since last year, when they immigrated from Guanajuato in northern Mexico to Painesville for work. Nathaly is behind in learning English, but is trying to get caught up, Dahlberg said, translating for Cordova and Medina
"She should be in second grade. But because they had to go through this process with immigration and the pandemic she got behind. And so she's actually in first grade," Dahlberg shared on behalf of Nathaly's parents.
The growing number of English learners like Nathaly has created challenges for the school district. Superintendent Josh Englehart says students are entering school behind, and the pandemic didn’t help; plus, absent students are a big problem. Painesville schools' chronic absenteeism rate - defined by the Ohio Department of Education as a student missing more than 10% of the school year - was 39% in 2023, compared to about 22% overall in the state.
"We have a lot of families who every winter return to Mexico with the kids," he said. "So that that's where that kind of transiency challenge really comes in . A lot of times annual trips in the middle of the school year out of the country."
Dahlberg said HOLA Ohio has partnered with the school district to create a "four-star" parent committee to start sharing facts with parents about how harmful it can be to students' academic progress to miss even a small amount of school. Englehart said the school district has actually seen a decrease in the number of students absent since then, and several other interventions have been put into place.
On the academics front, the district is doing what it can: translating communications to parents; assigning bilingual staff at each school; hiring English as a second language-certified teachers; and providing intensive language instruction several times a week, when students aren’t doing core subjects like math.
But that alone isn’t enough, Englehart says. The school district needs the partnership with HOLA Ohio.
"What Veronica provides us is, she has trusting relationships with so many people in the community," he explained. "She can have very frank and direct conversations with families in the community without insensitivity or offense. And she can turn people out (to events)."
Back at the community center, as kids’ attention starts to wander, the Legos and other toys come out. That’s Daniel’s favorite time; he’s in first grade, and likes to play with K’NEX. He says he likes science and math – which the games are meant to encourage - but when he grows up, he wants to be a barber. Does he think he'll use science and math as a barber?
"Yes, yes, yes," he said.
"Because I'm smart and... I'm beautiful," he explained.
HOLA Ohio does much more than host the weekly math tutoring nights. On top of helping with immigration cases and other advocacy for recent arrivals to Painesville, the nonprofit has a high school program for Hispanic girls in STEM - science, technology, engineering and math, a summer reading program and helps people get their driver's licenses.
Veronica Dahlberg, the executive director, says assessments are showing students starting to improve after the tutoring. But the stakes are high; test scores, graduation rates and other markers on the Ohio State Report Card remain stubbornly low for the district, which serves a large number of students in poverty across all races.
"We have the lion's share of the poverty and economic disadvantage in the county," Superintendent Englehart said. "We're the center of that in the county. And so we have always had massive gaps to close with our kids."
Meanwhile, HOLA Ohio can only do so much. It's hard to find enough volunteers at times, especially when - on an especially busy tutoring night - 60-plus students could show up. And funding is limited, Dahlberg said, with HOLA Ohio relying on community partners to donate money to cover the cost of food, for example. Finally, busy parents sometimes need prompting to bring their kids in. That's at least something that Dahlberg said the students don’t need.
"I know the kids love to come," she said. "The parents tell me, 'they say, is it Wednesday?' One kid, (his) mom told me that he doesn't want to go to school anymore. He wants to go to school at HOLA. Oh, I thought that was so cute."
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