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Showing posts from August, 2008

An AHA! abacus moment in the life of a preschooler

I've been doing math lessons with my 3-year old using the 100-bead abacus. Usually we do a few problems where she tells me a number to make and I tell her a number to make, back and forth. Today she asked me to make 51, and I asked her to make 68. These went smoothly since she's getting pretty good at this now.


Then we did a few "more than" problems. I said, "Let's say your sister has 5 cookies and you have one more than her. How many do you have?" This is a new concept to her so we need to do it slowly and carefully with the abacus: first make her sister's cookies, then let her have the same amount, then give her one more.

Then we do a few subtraction problems such as 7 − 4. She moved 7 beads, then "took away" or moved the other way 4 beads, and how many were left? 3 beads. I showed her also 50 − 10 = 40.

She started doing her own problem, "Let's do 9 − ..." and while she was thinking, I quickly proposed "... minus nine&qu…

Are these really parallelograms - answers

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These are answers to my earlier post where I asked if certain figures necessarily are parallelograms.

The question was: Does the given information in each diagram guarantee that each is a parallelogram?

Figure 1:
This one you can't get around; it ends up being a parallelogram, actually a dandy rhombus. Let's prove it. You can notice it has lots of sides of the same length. If we draw a diagonal, we get two triangles with all kinds of same sides:

The two triangles ABD and BCD end up having all three sides the same. So by the SSS triangle congruence theorem, they are congruent triangles. Hence, their corresponding angles are the same.

I've marked the corresponding angles with the same colors. Actually the triangles are even isosceles so the blue and purple angles are even congruent... but we don't need that fact.

To prove ABCD is a parallelogram, we need to prove its two sides are parallel. And for that, it's often handy to use the corresponding angle theorem: if correspon…

Multiplication vs. addition once more

Keith Devlin has published another column along the lines of multiplication not being repeated addition. I feel quite honored that he mentions THIS blog in his column (scroll down to the end), referring to what I wrote about the issue.

This time he expounds on research results. The research clearly shows that thinking of multiplication as repeated addition hinders students' further understanding of mathematics. It can lead to the misconception that multiplication always makes things bigger. Children need to acquire multiplicative reasoning, which is different from additive reasoning. And so on. Go read it yourself.

Bar diagram problem

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This was asked of me as of today:
Please solve this using the bar/block diagram method. My friends and I are stumped....

Desmond had 480 more oranges than pears. After selling half of his oranges and half of his pears, he had four times as many oranges as pears left. Find the number of pears he had at first.
Thank you!
This problem is from a Primary School Leaving Examination (PSLE) paper. PSLE is the final examination for primary school students in Singapore. So, you would expect to see these kind of problems in Singapore Math.

My first attempt for solving this was like this:

It shows the difference being 480. The red lines are halving the quantities of pears and oranges.

But I quickly noticed this was way off. The amount of pears needed to be way less than the amount of oranges.

My second attempt was like this:
It was a little better, but half the pears looked more like 1/3 of the half the oranges. So the amount of pears needed to still be less.

This is the final diagram that solves the probl…

Are these really parallelograms?

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Continuing with the idea in my post about Squares that aren't squares?, let's look at the following "parallelograms".

The question is the same:

Does the given information in each diagram guarantee that each is a parallelogram?

If you don't think so, your mission is to draw a quadrilateral with the given information but that clearly does NOT look like a parallelogram.

Figure 1:


Figure 2:


Figure 3:


Figure 4:


Again, these problems let students practice logical reasoning, and also learn about parallelograms, of course. See answers here.

Squares that aren't squares?

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Updated with solutions!

Today I want to highlight a square problem I saw at MathNotations. I hope Dave Marain doesn't mind me showing this picture and problem on my blog... I have no problem acknowledging it's from his blog. I COULD just tell you all to "go read it at Dave's blog....

BUT I don't feel that's the best way, IF I want you to think about this. I can just guess that most of the folks would feel too lazy to click on the link and go read it there (would you?). So I want to show it here.


Figures not drawn to scale! And this is important!

Now here's the question:
Does the given information in each diagram guarantee that each is a square?

If you don't think so, your mission is to draw a quadrilateral with the given information but that clearly does NOT look like a square.
The IDEA is to make our students THINK LOGICALLY, or practice their deductive reasoning skills. A great little problem.

The answers:


Figure 1 is not necessarily a square. The upper left…

Multiplication as many groups of the same size

It's been very good and educational for me to refine my thinking on multiplication vs. addition by reading some recent posts around the blogosphere, especially What's wrong with repeated addition by Denise and Devlin's Right Angle Finale at Text Savvy.

I feel that on some blogs people aren't even exactly talking about the same thing. The subjet we're dealing with - is multiplication repeated addition or not? - is subtle. Some people talk about how to define it - it is defined in some systems as repeated addition, and they feel that closes the issue.

BUT, I tend to agree with what Denise wrote: multiplication is a different operation from addition and somehow we need to get students to view it that way. I've always known that; I've never thought anything different. But yet how we present things to children is not always easy; we may understand the idea but not able to convey it right. Talking about multiplication as repeated addition MAY indeed leave the im…

Presentation at HOTM Conference

I just wanted to post a link to my presentation outline that I had yesterday morning at Heart of the Matter Virtual Homeschool Conference, if any of the participants (or non-participants) want to see it and see the links I gathered for it.

The Conference software didn't quite allow me to follow all those links during the talk so there were a few glitches. But I hope it was beneficial to those who listened. I was happy and excited for the opportunity to give the talk, though really nervous too! After all it was the first time for me to do such...

The idea of an online conference or virtual conference is just really neat! I was listening to the other speakers, and could get off any time to eat or do chores or whatever, then come back at my convenience. I got some ideas concerning scrapbooking your homeschool life, for example.

Icosahedron from picnic supplies

This was such a fun video to watch! This guy makes an icosahedron from plastic plates, and then from plastic cups - and one more from just plastic and duct tape.

It can serve as a fun summer math project for kids who love explorations, cutting, gluing, building, and that sort of stuff!

He is hilarious! See the video:


Video from Makezine.com

Read the instructions for the icosahedrons

Find out what is an icosahedron