About Functions

Can you guide me how to solve Question 7b?

This is from iGCSE past examination paper. I could not even start.

Given that f(x) = 10^x.
(a) Calculate f(0.5).
(b) Write down the value of f^-1(1).

Usman

f^-1(x) is the inverse of function f.
Given that function f(x) = 10^x
So, f(1) = 10^1 = 10
f(2) = 10^2 = (10)(10) = 100 and so on
I think you can figure out 10^(0.5) ... it is the square root of ten, isn't it?

Let's get to 7(b), which is the one you wanted help in.

First you need to know the meaning of inverse of a function.
If a function is 2x (doubling a number) then its inverse is (1/2)x i.e. halving the number.

Given that f(x) = 10^x, the second part is asking you what is the value of x when f(x) = 1.
The answer is x = 0 because 10^0 = 1.

(Note: I use ^ to mean to the power of)

You may want to review the idea of inverse function.

About Range

I am a fourth grade teacher and we are currently working with mean, median, mode and range.  I received a parent email asking to clarify the definition of range.  Her email is as follows:

“ A question came up during homework on the definition of range. Apparently the kids' textbook says that the "range is the difference between the least number and the greatest number".  Range = greatest value - least value. That is not the correct definition. Range is the least number to the greatest. For example, if the test scores in one class range are between 90 to 100, and another class between 60-70. The textbook definition would say the range for scores in both classes is 10, which obvious does not make sense. (I think their textbook is defining what is known as the span of the data but that is rarely used.) A number of parents were puzzled about it but we don't know if this needs to be corrected or at this level the textbook definition should stand. Would you please clarify.”

Am I correct that finding the range does require students to subtract the greatest from the least amount in the data set?  I am not sure how to respond to her question.

Jillian, New York

Definition of Range: 

There are many places one can check on the definition - wikipedia quoted reliable mathematics / statistics textbooks. This is another source for a definition.

You can send the parent some of these links. 

We are often interested about the 'average' in a data set as well how the data distributes itself around the average. Examples of average include mean and median. examples of a measure of this distribution includes range and standard deviation.

Range is the size of the smallest interval that contains all the data and tells us about statistical dispersion.

I tried checking what the parent referred to as "span of the data" but could not find any entry on the internet. Apparently, it is not a conventional term. Range is a more conventional term to describe the idea under discussion. 

Question on Division Bar Model

Trish from Hawaii asked a question about setting up the bar model in a division type word problem.

Question by Trish (Hawaii) from Jimmy Keng

Of course it is not necessary to solve every problem using the bar model.

Spiral Curriculum

Good day Dr. Yeap! I attended the seminar last May 21-24 at SM Megall (see photo). Thank you for sharing your expertise and time with us. I just want to ask you about spiral progression. How do you apply it in Math? Can you give an example? Cely

Singapore mathematics curriculum emphasizes the spiral approach based on Jerome Bruner's explanation on spiral curriculum. The idea of the spiral curriculum, according to Bruner - 'A curriculum as it develops should revisit this basic ideas repeatedly, building upon them until the student has grasped the full formal apparatus that goes with them.' Most people will not miss the idea of 'repeatedly' but may miss the subtle notion of 'building upon them' and 'until the student has grasped the full formal apparatus' of the target concept. In Singapore curriculum, addition is taught four times in Grade 1 (this is a core idea and they are new to it) - addition with 10, within 20, within 40 and within 100. Students get to revisit the idea of addition repeatedly but each time building on the strategies that they already had. When they add with 10, they count all and count on, perhaps with the use of concrete objects and drawings. Later, in addition within 20, they learn to make ten before adding, effectively acquiring the notion of place value. Later they progress to more formal approaches such as adding ones and adding tens in the formal algorithm. Thus, it is not mere review of materials. It involves extension. In a similar way, multiplication of whole numbers is taught in grades one through four; addition and subtraction of fractions is taught in grades two through five; area of plane figures is taught in grades three through seven; solving equations is taught in grades seven through nine. As a result, in Singapore, Algebra is taught across grade levels in high school (grades seven through twelve). Thus, we do not have the practice of teaching Algebra, Geometry etc separately. They are all under the subject of mathematics. There is geometry in all grade levels.

Question on Retention

Question - How do we help students retain what they have learned.

ATeacher in Hawaii

Answer

A good program arranges the topics in a certain way for various purposes. To help students retain, one of the ways is to arrange topics in such as a way that students have many opportunities to revisit a particular key concept or skill.

Let's use the example of equivalent fractions. After it is learned in Grade 3, the chapters that follow it are adding unlike fractions and subtracting unlike fractions (but limited to cases such as a third plus a sixth or three tenths subtract a fifth where it is necessary to rename only one of the fractions to make both like fractions). The two weeks or so of constant writing fractions as equivalent ones as students add unlike fractions help them consolidate the skill learned. It is critical that a new skill is well consolidated before students leave them behind.

The next grade level when student said such fractions but for cases where the sum exceeds 1, students get to review finding equivalent fractions. In grade five, when students are dividing say a half by three, one of the methods involves renaming one half as three sixths before proceeding to divide the three parts into three. Using this method, three fourths divided by three can be done straight away but to divide three fourths by two requires renaming three fourths as six eighths. The opportunity to review equivalent fraction in a different context enhance the retention of the skill of finding equivalent fractions.

Two principles discussed here - ample consolidation after a new skill is learned and review but not a mere repetition but done in other / more challenging contexts.

Singapore curriculum is arranged for this to happen. Good textbook authors arrange the topics for this to happen.

A Bunch of Questions

Question
How can Singapore Math be taught to children who are at different learning levels? Answer
We just completed a two-day professional development on how to do differentiated instruction using Singapore Math in White Plains with 60 teachers. For struggling learners the concrete experience before pictorial representation helps them. For advanced learners, tasks can be easily extended to engage them in higher-order thinking. Singapore Math is well-known for helping average learners reach high levels of achievement. The last piece of information is what emerges from TIMSS and PISA where many of our average learners are performing at Advanced level in TIMSS or Levels 5 and 6 at PISA.

Question
How does Singapore Math help children who have difficulty learning math?
Answer
The use of visuals helps. In Singapore Math we use the CPA (concrete-pictorial-abstract) Approach based on Jerome Bruner's work. Students are taught an abstract concept via concrete expreinecs and the use of pictorial representations.

Question
How does Singapore Math help children with more advanced math skills? Singapore Math is based on the idea of using mathematics as a vehicle for the development and improvement of a person's intellectual competencies. Students with advanced skills get to work with more complex problem that requires deeper thinking (this is in the program). They also get to develop skills such visualization and ability to see patterns.

Question
What training do teachers/school administrators need in order to introduce Singapore Math in their curriculum?
Answer
Marshall Cavendish Institute offers a range of professional development courses to help teachers teach the program. MCI's Experiencing Singapore Math is a one-day executive program for administrators who are keen to implement the program in their schools. The publisher HMH has ran this program in Chicago, Nashville, Alabama, Scottsdale, and recently in Neward and East Brunswick in NJ.

Question
What grade levels seem to show the greatest impact for Singapore Math?
Answer
I do not know this but you may refer to various research done at various sites. Based on my professional experience, best results come when the kids are introduced to it at K-2.

Question about Students Not Meeting the Goals

Question:
What if a student is clearly not ready to move on to the next level at the end of the school year despite concerted efforts to assist him/her? What do you folks do with these students (grade K-6)?

Malama pono (Take care),
Teacher in a oublic charter school in Hawai'i

Answer:
coming soon