The Tutor is set to work with students ages 7 and up, with some exposure to the mathematics typically presented in a K-2 curriculum. To be specific, the Tutor presumes the following.
The student has...
- had exposure to words one, ten, hundred, thousand, ten thousand, one-hundred thousand, and a million. They need not have a deep intuitive understanding of these counts (such as being aware that a thousand thousands matches a million), nor have a sense of what exactly constitutes a billion.
- had exposure to the idea that ten 1s make 10, ten 10s make 100, ten 100s make 1000.
- had first exposure to the idea that a number like 8,672 represents eight 1000s, six 100s, seven 10s, and two 1s and knows the English words for 10, 20, …, 90. (They are quirky!)
Consequently, we presume students...
- could count to a million (if they really wanted to!).
We also presume students...
- can count by 2s, 5s, 10s, 100s, 1000s.
- have a strategy for adding two single-digit numbers or a multi-digit number and a single digit number (as strategy like “counting on”). They might not be speedy in this.
- can add multiples of 10 or multiples of 100 together.
- have an understanding of “take away” (aka subtraction) and have a strategy for subtracting a single-digit number from another number (likely counting backwards).
- recognize that a subtraction problem can be solved by finding an unknown addend in an addition problem. (e.g. The value of “10 - 7” can be found by asking “What adds to 7 to give 10?”)
- can add or take away 10 or 100 from a number with ease.
- can decompose numbers. (Rewrite 10 as 3+7, for instance ,and 200 as 60 + 40 + 100.)
- can solve basic problems of each of these forms: a+b=?, a+?=b, ?+a=b, a-b=?, a-?=b, ?-a=b and simple word problems that lead to such equations.
- have explicit awareness that numbers are ordered: that 7 is less than 9, and if a number M is “N plus some more,” then M is bigger than N, for instance. (We don’t presume students know the notation < , >, etc.)
- have seen numbers being represented, in order, on a number line.
- have had some first exposure to the idea that repeated addition among the counting numbers is called multiplication.
- have seen the symbols + , -, for addition, subtraction, multiplication, along with = for equals at some cursory level.
And finally, we presume students have a general grasp of:
- basic shapes
- and that reorienting shapes does not change their fundamental characteristics
We shall discuss many of the topics listed above, but we will do so with the assumption that this is a return to a previously seen concept. We’ll come to it in a new sophisticated light and new mathematical sophistication. We will be articulating nuances about the topic that students don’t normally realize, and pre-empting common misconceptions that often lie undetected.
We don’t presume speedy fluency with “math facts.” But we will certainly offer fluency experiences for students to enjoy and to engage in practice.