Ask Marian

I recently made such a list for the vice-principal of an elementary school. Here’s what I included on my list of manipulatives as must-haves and desirables for classrooms from Kindergarten to Grade 8:

Must-Haves:
Linking cubes
Objects to count (e.g., buttons, pebbles)
Square tiles (they come in four colours)
Transparent mirrors
Fraction strips
Counting rods (also called Cuisenaire rods)
Number paths
Pattern blocks
Base ten blocks
Two-colour counters (different colour on each side)
Number cubes/number rollers with various numbers of sides (e.g., 1–10, 1–6)
Pan balances
3-D solids (plastic or wooden)
Mini whiteboards
100-charts (large format)

Desirables:
Number racks (such as Rekenreks)
Algebra tiles
Geoboards
Plastic strips for making and measuring angles
Plastic links

Technically, fractions and ratios are different things.

A fraction is a number on a number line, while a ratio is a comparison of two numbers.

That said, the size of the fraction is completely determined by the ratio of its numerator to its denominator, and so ratios and fractions are closely linked.

We put (frac{2}{4}) on a number line in the position we do because the ratio of the numerator to the denominator is 1:2, so we go halfway between 0 and 1 on the number line.

We put (frac{2}{3}) where we do because the ratio of the numerator to the denominator is 2:3, so we split the section from 0 to 1 into 3 equal parts and move forward only 2 of those parts.

It’s pretty tricky. You could press the number π on a calculator and it certainly would look like the decimal digits do not repeat. But it’s pretty hard to be sure.

It turns out that to prove that the digits do not repeat is beyond the math that almost any student in intermediate grades could handle.

I think drawing students’ attention to the decimals on a calculator is the best bet, although we still have to reinforce the idea that we can’t be sure since we see only a small portion of the digits in what is an infinite decimal.