An example of figural representation is given in Figure 1. The contextualization of the items was introduced to make sure that children based their answer on both columns of the tables.
For the number category, there were four types of questions. The first question was a comparison of fractions. Pupils had to decide which of two fractions represented the larger quantity.
There were fractions with the same numerator e. In the second question, pupils were asked put fractions in ascending order. This question also involved improper fractions and natural numbers.
The third question involved finding a fraction between two given fractions e. Fractions with common denominators, common numerators, and no common components were included. For the fourth question, pupils were asked to place a fraction or the unit on a graduated number line e. The given references were always 0 and another fraction. We assessed the following procedures: addition and subtraction with or without the same denominator; multiplication of fractions; multiplication of a fraction by an integer; and simplification of fractions.
Division of fractions was not included as it is not part of the official curriculum. Descriptive statistics are reported for each category of fractions part-whole, proportion, numbers, operations, and simplification.
Mean scores and standard deviations are always expressed in percentage. As can be seen in Table 1 , children performed better for questions about proportion and part-whole than for questions about the other categories. There were still major difficulties in Grade 6 for the part-whole category.
Indeed, even in Grade 6, the percentage of correct responses was still far from ceiling performance. Children were capable of resolving questions on proportional reasoning from Grade 4. The main observed errors were linked to additive reasoning. Children got the lower scores in Grade 4 for arithmetic operations. This was not surprising as learning about operations on fractions usually start in Grade 5.
Table 1. Mean percentage of correct responses and standard deviation for each category in Grade 4—6. A correlation analysis was run to assess the relations between conceptual part of a whole, proportion and numbers and procedural categories operations and simplification.
The correlation analysis revealed that conceptual categories correlated significantly with each other see Table 2. They also correlated positively with procedural categories. We ran an ANOVA for repeated measures with category as a within-subjects factor part-whole; proportion; number; operations; simplification and grade as a between-subjects factor.
Figure 2. The top two panels show the interaction between grade and correct response rates for each category A , and between grade and each type of knowledge B.
The bottom two panels show dendrograms depicting the results of a single linkage hierarchical clustering of each category based on Euclidian distances for Grade 4 C and Grades 5 and 6 D. We ran another ANOVA for repeated measures on the type of knowledge conceptual and procedural with grade as a between-subjects factor. We also ran cluster analyses to ensure that our categories reflected conceptual and procedural knowledge. Since two patterns appeared in the results, we ran two separate cluster analyses: one analysis for Grade 4 and one analysis for Grades 5 and 6.
We ran neighbor-joining analyses single linkage method to see if our categories formed natural clusters that could be labeled according to a type of knowledge. These analyses provide a tree-structured graph i. The dendrogram indicates at what level of similarity any two clusters were joined.
It was constructed using neighbor-joining algorithm based on Euclidian distances. Both for Grade 4 and for Grades 5 and 6, the dendrograms clustered the categories into two distinct groups that correspond to our two types of knowledge, i. Part-whole, number and proportion were the most similar and correspond to our conceptual categories, whereas operations and simplification can be combined in a different cluster, that is our procedural categories.
Different variables were involved in this question. Table 3. Mean percentage and standard deviation for the question: Draw a representation of the given fraction. Despite potential graphic difficulties, pupils mostly divided a common continuous shape circle or square, see Figure 3. Figure 3. Illustration of the most common answer when pupils were asked to draw a representation of a given fraction.
Performance was better for continuous quantities. Mean scores per grade are given in Table 4. Table 4. As seen in Table 1 , performance for proportion items was better than in other categories. Percentage of correct responses showed a clear difference between three groups of items. In the first group of items, there were 3 number lines for which pupils only had to count the number of graduations corresponding to numerators to succeed e.
For these items, they could only process the numerator and ignore the denominator. In the second group of items, there were two number lines on which pupils had to place 1 e. The third group of items involved equivalent fractions e. These errors are consistent with the errors observed in the number line task. Children struggled with the relation between fractions and the unit. Pupils had to choose which of two fractions was larger. Performance for addition and subtraction with same denominators was better than for addition and subtraction with different denominators see Table 5.
This is not surprising as addition and subtraction with different denominators are not yet part of the program in Grade 4. But the procedure to find the lowest common denominator seems to pose problems in Grade 5 and 6. The most common error was based on the natural number bias, that is, adding or subtracting numerators and denominators as if there were natural numbers e. Surprisingly, performance for multiplication of fractions was better in Grade 4 than in Grade 5.
Table 5. Mean percentage of correct responses and standard deviation for each type of operations in Grade 4—6. As can be seen in Table 6 , performance in the simplification task was better for fractions that could be divided by 2 e.
Table 6. Mean percentage of correct responses and standard deviation for the simplification task in each grade. In this study, we investigated the difficulties encountered by primary school children when learning fractions. One of the main goals of this study was to clarify the relationships between conceptual and procedural understanding of fractions.
In order to do so, a test was administered in Grade 4—6 in classes of the French Community of Belgium. Globally, the results showed large differences between categories. Pupils seemed to master the part-whole concept, whereas numbers and operations posed tremendous problems. Some conceptual meanings, such as numbers, were less used in primary school classes.
Part-whole seems to be a concept that is widely used in the classrooms. However, they seem to have a stereotypic representation of fractions. Indeed, when they were asked to represent a given fraction, they mostly used a circle or a square, even when drawing collections could have been easier e. Moreover, when asked to select a figure representing a certain fraction, they performed better for continuous than discrete quantities. Pupils performed well with proportion items.
These results contrast with textbooks and lessons given by teachers. In fact, the connection between proportions and fractions is rarely made in textbooks and formal lessons, even if some aspects of fractions are based upon proportional reasoning e. In the proportion category, most errors were linked to additive reasoning. In this case, children built their answer on only a subset of the given information and they applied additive strategies where multiplicative strategies should be used.
Mistakes linked to additive reasoning are commonly reported during early stages of children's understanding of proportional reasoning Lesh et al. This kind of mistakes was common in Grade 4, but could still be observed in Grade 6. Pupils performed poorly in the numerical category. Even if children are trained to deal with number lines from grade 4, results showed major difficulties when they were asked to place a fraction on a graduated number line.
They do not seem to have an appropriate representation of the quantities of fractions. Other studies have reported that many pupils experience difficulties when asked to locate a fraction on a number line.
Pupils often view the whole number line, irrespective of its magnitude as a single unit instead of a scale Ni, When they are asked to place a fraction between 0 and 1, pupils often place fractions disregarding any other reference point or known fractions. Pearn and Stephens pointed out that the incorrect location of fractions could also be the consequence of a lack of accuracy when dividing segments.
The lack of accuracy in children's mental representations of the magnitude of fractions seems to be confirmed by the weak percentage of correct response for questions involving sorting out a range of fractions in ascending order. Furthermore, mean percentage of correct responses for comparison of fractions were very low for fractions with common numerators and fractions no common components.
When fractions share the same denominator e. In this case, pupils could only compare the numerators in order to choose the larger fraction. When fractions share the same numerator, the global magnitude of fractions is incongruent with the magnitude of denominators. Thus, pupils might not take the incongruity into account and their judgment might have been influenced by the whole number bias Ni and Zhou, For fractions with no common components, pupils probably only compared numerators and denominators separately.
This strategy led to larger error rates. Focusing now on operations, children performed well in addition and subtraction of fractions with the same denominator, while performance dropped dramatically in addition and subtraction of fractions with different denominators. The most common errors were dictated by the whole number bias Ni and Zhou, Surprisingly, results were poorer for items involving the multiplication of an integer by a fraction, than for multiplication of two fractions.
In the last case, pupils could successfully apply procedures based on natural numbers knowledge, which would explain higher percentage of correct response. Another surprising result was the better performance in Grade 4 than Grade 5 when children were asked to multiply an integer by a fraction.
There might be a contamination of procedures applied to addition and subtraction with different denominators learnt in Grade 5. Results showed massive familiarity effects in every category. We do not know precisely when children start to quantify continuous quantities in informal contexts. Their results suggest that using the concept of half would be the first step in relationships used by children to quantify fractions.
They had to judge if they would receive the same amount of chocolate bars in both parties, and if not, in which party they would get more chocolate bars. Children had ceiling performance when they could use half as a reference. In the condition where they could not use half as a reference, only 8-year-olds had performance above chance.
Desli also showed the importance of the concept of half in the construction of fractions quantifications. In a recent study using a fraction-based judgment task, Mazzocco et al. In this section, we'll look at what these questions have in common and give examples of each type.
Some of the reasons why the hardest math questions are the hardest math questions is because they:. Secret to success: Think of what applicable math you could use to solve the problem, do one step at a time, and try each technique until you find one that works! We must solve this problem in steps doing several averages to unlock the rest of the answers in a domino effect. This can get confusing, especially if you're stressed or running out of time. Secret to success: Take it slow, take it step by step, and double-check your work so you don't make mistakes!
For example, many students are less familiar with functions than they are with fractions and percentages, so most function questions are considered "high difficulty" problems. If you don't know your way around functions , this would be a tricky problem. Secret to success: Review math concepts that you don't have as much familiarity with such as functions. We suggest using our great free SAT Math review guides. It can be difficult to figure out exactly what some questions are asking , much less figure out how to solve them.
This is especially true when the question is located at the end of the section, and you are running out of time. Because this question provides so much information without a diagram, it can be difficult to puzzle through in the limited time allowed.
Secret to success: Take your time, analyze what is being asked of you, and draw a diagram if it's helpful to you. With so many different variables in play, it is quite easy to get confused. Secret to success: Take your time, analyze what is being asked of you, and consider if plugging in numbers is a good strategy to solve the problem it wouldn't be for the question above, but would be for many other SAT variable questions.
The SAT is a marathon and the better prepared you are for it, the better you'll feel on test day. Knowing how to handle the hardest questions the test can throw at you will make taking the real SAT seem a lot less daunting. If you felt that these questions were easy, make sure not underestimate the effect of adrenaline and fatigue on your ability to solve problems.
As you continue to study, always adhere to the proper timing guidelines and try to take full tests whenever possible. This is the best way to recreate the actual testing environment so that you can prepare for the real deal. If you felt these questions were challenging, be sure to strengthen your math knowledge by checking out our individual math topic guides for the SAT. There, you'll see more detailed explanations of the topics in question as well as more detailed answer breakdowns.
Felt that these questions were harder than you were expecting? Take a look at all the topics covered in the SAT math section and then note which sections were particular difficulty for you. Next, take a gander at our individual math guides to help you shore up any of those weak areas. Running out of time on the SAT math section? Our guide will help you beat the clock and maximize your score.
Aiming for a perfect score? Check out our guide on how to get a perfect on the SAT math section , written by a perfect-scorer. Check out our best-in-class online SAT prep classes. We guarantee your money back if you don't improve your SAT score by points or more.
Our classes are entirely online, and they're taught by SAT experts. The more positively you receive all answers that are given, the more students will continue to think and try. There are many ways to ensure that wrong answers and misconceptions are corrected, and if one student has the wrong idea, you can be sure that many more have as well. You could try the following:. Value all responses by listening carefully and asking the student to explain further.
If you ask for further explanation for all answers, right or wrong, students will often correct any mistakes for themselves, you will develop a thinking classroom and you will really know what learning your students have done and how to proceed. If wrong answers result in humiliation or punishment, then your students will stop trying for fear of further embarrassment or ridicule. Right answers should be rewarded with follow-up questions that extend the knowledge and provide students with an opportunity to engage with the teacher.
You can do this by asking for:. Helping students to think more deeply about and therefore improve the quality of their answer is a crucial part of your role. The following skills will help students achieve more:. As a teacher, you need to ask questions that inspire and challenge if you are to generate interesting and inventive answers from your students. You need to give them time to think and you will be amazed how much your students know and how well you can help them progress their learning.
Remember, questioning is not about what the teacher knows, but about what the students know. It is important to remember that you should never answer your own questions! After all, if the students know you will give them the answers after a few seconds of silence, what is their incentive to answer? Every effort has been made to contact copyright owners. If any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangements at the first opportunity.
Video including video stills : thanks are extended to the teacher educators, headteachers, teachers and students across India who worked with The Open University in the productions. Printable page generated Friday, 12 Nov , Use 'Print preview' to check the number of pages and printer settings.
Print functionality varies between browsers. Printable page generated Friday, 12 Nov , TI-AIE: Asking questions that challenge thinking: fractions What this unit is about In this unit you will think about how to introduce fractions to your students. What you can learn in this unit How to ask effective questions that are interesting and challenging. Some ideas to help your students construct their own understanding of fractions.
Some ideas to help your students talk about fractions. In her research, Nunes found that primary school students already have insights into fractions when solving division problems: They understand the relative nature of fractions: if one student gets half of a big cake and the other gets half of a small one, they do not receive the same amount.
Talking fractions: using the language Encouraging the students to talk about fractions and use the vocabulary will help them understand some of the difficult vocabulary associated with fractions. The first activity is for you to think about issues of learning fractions in your classroom. Activity 1: Thinking about your students learning fractions Think about what your students need to know about fractions, and make some notes on the different ideas.
If you have a multigrade class, you will need to think about what different students need to know about fractions: how to find out a fraction of a quantity what fraction one quantity is of another how to add fractions together. Activity 2: Physically representing fractions Preparation First create a space, and ask eight students to come to the front of the class or somewhere where the rest of the class can see them.
The activity Ask your students to arrange themselves into a rectangle. Ask someone else to divide the group into half. Reform the rectangle, then ask another student to divide the group in half in a different way.
Ask the students what is the same and what is different about the new half of the group. Now ask another student to divide the eight students into quarters fourths.
Again ask whether there is a different way to do this division, and what is the same and what is different about the new way of dividing into quarters. Now change the number of students and go through the process above again. It may be that dividing into quarters is difficult but depending on the chosen number, continue asking for one divided by two , one divided by four , one divided by three and so on, until a fraction that cannot be done is reached.
Ask the students why you cannot find that fraction of these students. Dividing one student into bits is not allowed! Ask the students to work in groups of You could appoint a leader in each group to note down ideas if the class does not split evenly into groups of Ask them to work out all the fractions they can divide 12 students into.
Video: Using questioning to promote thinking. Case Study 1: Mrs Rawool reflects on using Activity 1 This is the account of a teacher who tried Activity 1 with his elementary students.
Reflecting on your teaching practice When you do such an exercise with your class, reflect afterwards on what went well and what went less well. Pause for thought Good questions to trigger such reflection are: How did it go with your class?
What responses from students were unexpected? Did you feel you had to intervene at any point? What points did you feel you had to reinforce? Did you modify the task in any way? If so, what was your reasoning for this? Remember to use what you know about manipulating equations—such as whatever you do to one side you must do to the other—to handle these problems.
Category: Problem Solving and Data Analysis—standard deviation, range, analyzing graphical data. Standard deviation is a measure of spread in a data set, specifically how far the points in the data set are from the mean value. A larger standard deviation means that the points in the data set are more spread out from the mean value, and a smaller one means that the points in the data set are close to the mean value.
Range refers to the difference between the highest and lowest values in a data set. The SAT may require you to calculate the range for a data set.
In contrast, the second data set is more spread out, so we can conclude that the standard deviation of the first set is smaller than the standard deviation of the second. We can eliminate choices A and C. We can subtract the highest and lowest values for each set to find the range. The ranges of each set are equivalent. This leaves us with choice D as our answer. Category: Passport to Advanced Math—quadratic and algebraic functions and their graphs. We can solve this using substitution.
Substitute the second equation into the first and expand the quadratic:. Next, move the x from the left-hand side over to the right. The roots of this quadratic will give us the solution to the system of equations above. Since the roots of the above quadratic will give us the solution to the system of equations above, we can use the discriminant of the quadratic formula to find out how many solutions there are.
If the discriminant is positive, there are 2 solutions; if the discriminant is 0, there is 1 real solution or a repeated solution ; if the discriminant is negative, there are no real solutions. This is a positive number, which means there are 2 solutions to the system of equations.
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