Most times, when we start on a topic of study, we find it very diffcult to draw a line under it and to move on to something else. I mean, when does one decide that he/she has learnt enough to move on to the next topic? I thought this week we would do a little bit of introduction to fractions then move to the next topic, but we ended up spending the entire week on it as we found more things to learn in fractions, e.g. applying the four operations to fractions, and converting between mixed numbers and improper fractions.
How did we end up doing more and more without intending to? Well, it all came about because I thought it would be a good idea to have a little review of the basics:
That was all I wanted to do. However, the last question in the clip above: Estimate 7/8 + 12/13 led Tiger to find out about conversions, which I explained using on a few pieces of paper. Not very sophicated, I know, but the most basic tools are sometimes the most effective.
Once he has understood my explanations, Tiger wanted to try out some activities to test his own knowledge so I let him test himself here, but not before he has looked through the revision section and the activity section.
We first started looking at fractions about two years ago but hadn't formally done much about it until this week.
I like to introduce maths in a visual, concrete way, so I took out two manipulatives that have been tucked away in the cupboards from Tiger's preschool days and let him play around with them for a bit:
The Equivalency Cubes can also be used to teach concepts of of percentages and decimals. The main purpose of them this week, however, is for Tiger to become familiar with the concept of equivalent fractions.
After Tiger has played with the cubes for a while, we started on our fraction activity. I gave them three sheets of papers, each printed with three circles to be divided into smaller fractions. For example, the first sheet consists of three circles that have been divided into halves by a line drawn through the centre of the circle. The idea is to have the first circle remained as halves, and have the student further divide the other two circles into fourths and eighths. The second set of circles are: thirds (already drawn), sixths, and twelves; the third set consists of: fifths (already drawn), tenths, and twentieths.
After the division of the circles into smaller fraction parts is done, Tiger cut them out and glued them onto a piece of construction paper (together with each circle's corresponding labels) to make "The Dangerous Fraction Garden", as Tiger puts it. This step is not strictly necessary to learn the maths concept, but we wanted to have a laugh so we did it.
We then proceeded to understand Fractions of a Set. Using the printed sheets of incremental stacks of 4x5 squares, we first determined the total of each set. Then, we talked about how much one-fourth of each set would be -- this is quite simple, given that the sets are designed in four rows each. Finally, from one-fourth of each set, Tiger could easily determine three-fourths of each set by multiplying by three.
When the colouring is done, we glued both sheets of paper together to have an overall view of all our three-fourths. It provides a clear visual sense of what three-quarters of an incremental total looks like:
While Tiger has labelled under each set its non-simplified three-quarters fraction (15/20, 30/40, 45/60, 60/80, 75/100, 90/120), I showed him how to simplify them by finding the common factor of both numerator and denominator, so that each fraction eventually reduces to 3/4.
Tiger then worked through some of the problems in the Fractions section at Khan Academy. We found the drills to be adequate for practice.
The word problems on fractions present an interesting way to apply what Tiger has learned so far, so that is well worth a try. When he had enough of drills, he played the Fractions and Coins Game.
Finally, we also learned not to get too hung up about getting perfect scores, especially when in France:
At the beginning of the school year, we noticed that Tiger has gotten the hang of single-digit multiplications up to nine. I credit his success at cracking the multiplication tables without any pain nor any need to memorise multiplication tables to the RightStart curriculum that we are using, since we don't use anything else for mathematics. In fact, we are only just 'officially' starting with multiplication in this programme, but I think the strong foundations in addition and the way the curriculum is written just works to help Tiger make the mathematical connections himself.
A knowledge of multiplication is the pre-requisite to starting fractions. Since this topic is not introduced in RightStart until the end of springtime next year (according to our schedule), I am using a few other resources to do the job. We are doing fractions alongside our main math curriculum, about once a week.
As an introduction, I let Tiger freely play with the Fraction Circles. Putting the circles back to their original shape by colour was too straightforward for Tiger, so he decided to see whether they could make full circles using different combinations of the fraction parts.
I wrote down all the combinations of the circles he made. This sheet of paper will be used later when he has learnt about equivalent fractions to verify whether the various combinations work towards a whole.
To keep Tiger's interest going and to introduce the concept of fractions gently, I had Tiger read the following book. The book is bright and colourful, and introduces the concept in an interesting way such that after reading it, Tiger asked to do fraction work, to which I gladly let him practise on the Modeling Fractions worksheets found here.
The worksheets above were a huge confidence booster for Tiger and he asked for more, so I whipped out Book 1 of Key to Fractions. Book 1 is all about concepts so it is very straightforward and Tiger has no problems with the workbook so far. It is comforting to see him learning this concept so easily but I intend for this part of maths to be supplementary to our main curriculm. I think we have more interesting things to do with our lives than to sit through multiple workbooks.
This post is linked up to several blog hops, where you can visit to see what other homeschoolers have been busy with.
We were on one of our family nature walks a few evenings ago when Tiger got through his multiplication facts, as well as negative numbers. This happened casually when Tortoise was playing a numbers game with him as we walked.
Based on our math curriculum, multiplication has not been formally introduced yet while negative numbers are done as the very last lesson of Level C, but we had spent the past two years (starting from Level A) working on additions, so that had formed the basis for Tiger to understand that both multiplication and negative numbers are not much more than addition "flipping forwards and backwards" (Tiger's words).
Tiger was very pleased that he was able to figure out the correct answers to the game that he wanted to play it over and over again the following day, so I got "Mum, ask me another multiplication question!" and "Mum, I know what's minus such-and-such number minus/add such-and-such number is!" over lunch, during break time, and while cooking. I guess that's his multiplication basics done.
Last night while playing chess, Tiger and Tortoise's conversation somehow turned from figuring out the number of squares on the chess board to division to binary numbers, then to fractions. Tiger understood binary numbers and how to work them out - which was a reinforcement on yet another way to apply multiplication facts. However, he seemed a little muddled about fractions but was very keen to learn it. I looked at our Level C book, and found that fraction does not get introduced until near the end of the book in Lesson 133.
While I continue to believe in RightStart's approach to build solid foundations in mathematical understanding, I do not want to hold Tiger back when he is showing a keen interest in a certain area. So, here I am, scrambling around to get hold of appropriate materials to teach fractions starting from next week, while figuring out how much time to take out of our schedule to tackle this new topic.
