*Quiz #1 as a Word Document:*

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*Question 1,* Fractions. It’s Payoff Time!

We’ve been having students ** draw **fractions

**for weeks now. Weeks! We are gradually increasing the difficulty of the questions, to cover addition, subtraction and multiplication of fractions. We assign about 1 problem a day. And… drumroll…90% got this problem right now. It involves multiplication and borrowing! Neither of which we’ve**

*only**taught.*We’ve just given students the

*tools*to make sense of fractions.

The 3rd example below is one of three students who are still confused. At this point, this can be remediated in one 15-minute session.

** Question 2, Patterns.** The first example below is correct. 30% of students got the answers correct up all the way up to the ‘nth’ figure. 65% solved up through Figure 9. (see 2nd example below). They can see (this is all about SEEING!) that the drawing is a

*square*plus one block. The square is one more than the figure number. Figure 2 contains 3 squared, Figure 3 contains a 4 squared, so Figure 9 contains a 10 squared. Then add the 1 extra block. Ten squared plus 1 = 101.

The last 2 examples are representative of the 35% of students who had trouble with this pattern. However, their work that shows effort and understanding; perhaps they have either counted wrong or drawn the figures wrong. We feel this is developmentally normal. Our goal now is to expose students to the thinking involved in observing and predicting a pattern. The abstract thinking involved in finding Figure ‘n’ is well beyond grade level, and is simply meant as a challenge for those who are ready for such abstract representation.

Since our grading system grades students at the three different levels (see blogs from Oct 9, 10 and 11), we are able to give students validating feedback at the highest level they can handle; their “Zone of Proximal Development”.

** Question 3, Area Model. ** All but 2 students got this correct. A few reached for their blocks to make sure (fine!). The second example below shows one of the students who got it wrong. This student drew it correctly, but is not counting the length of the sides correctly. She is confusing LENGTH (linear units, perimeter, rulers, string) with AREA (quadratic units, carpet squares, tiles, flats, paper squares). This is a very common and understandable mistake at this age (indeed it still confuses Algebra I students later, who cannot see the difference between

*x*and

*x squared*and try to add them to get

*x cubed!)*When remediating, we use uncooked spaghetti pieces to lay alongside the edges of the rectangle. This wonderfully illustrates how to count the length of the sides of the rectangle. (more about this in a couple weeks, when we use spaghetti lenghts to find decimal area!)

Challenge Questions

*Question #4.* Only 2 students got this one correct (see first example below for one of the correct answers), which is *fine.* What we like is seeing the *attempts* students made to figure it out, and show their thinking (examples 2, 3, and 4 below). (What is most difficult here is the realization that the terms are being *added*, not multiplied. You actually have THREE GROUPS of 3^{x}. That means 3 times 3^{x}. Or 3 times itself x times, and then times 3. So 3*3^{x} can be thought of as 3^{x+1}. ) This is what we think *rigor *means. Not moving as quickly as possible through concepts to be memorized, but engaging in questions with depth, questions that make us uncomfortable and curious and confused (see example 5 below!). We want even our fastest students to feel what struggle and confusion feel like — that’s what mathematicians do.

*Question #5.* Even though we’ve never taught the binary system, it’s shown up on so many homework assignments that students have figured it out, and find it fairly easy.