## Reflection: Connection to Prior Knowledge From Guess and Check to Creating Equations; From Equations to Inequalities - Section 2: Group Problem Solving: Vanessa's Raise and Ed's Book

The "Ed's Book" problem offers a fantastic opportunity to connect to prior knowledge.  First of all, let me say that this was a student's idea before it was my own - and that I wish I'd had the time to grab a picture of his work! - but I've recreated it here: Number Line for Eds Book.  These days, there will often be a student or two who come up with this idea, and if that doesn't happen I'll share it as an example.

What students see is that they can use a number line to represent this problem.  Earlier in the year, we did a lot of work with number lines, particularly with the placement of fractions on number lines, and this problem is a chance to apply that knowledge.

First, we create a number line that runs from 0 to 1, with 0 being the beginning of the book and 1 being the end (ie, 100% of the way through).  Next, we can place 1/2 and 2/3 accurately on the line.  If we notice that the "distance" between those is 84 pages, then we just have to figure out how much "100%" of the book must be.

From here, many students can solve the problem.  If they get stuck, there are two routes that I might take them on.  One is to notice that, in order to compare the fractions 1/2 and 2/3, they ought to have a common denominator.  If we notice that 1/2 = 3/6 and 2/3 = 4/6, then it follows that the 84 pages Ed says he's going to read must comprise 1/6 of the book.

The other way to think about it involves noticing that 1/2 is equidistant between 1/3 and 2/3 on the number line.  Kids can see this with a rough visual sketch or by thinking about each number in decimal form (.50 - .33 = .17 and .67 - .50 = .17).  This is very informal, but it can help students continue to grasp the meaning and the usefulness of fractions.  If we accept that, just like it is from 1/2 to 2/3, the distance between 1/3 and 1/2 is 84 pages, then the middle 1/3 of the book must be 168 pages.  And if 1/3 of the book is 168 pages, then the whole book must be 504 pages long.  Then, if all of that seems hard to believe, we can check to make sure that 504 pages fits into everything Ed told us originally.

Connection to Prior Knowledge: From Number Lines to Ed's Book

# From Guess and Check to Creating Equations; From Equations to Inequalities

Unit 4: Creating Linear Equations
Lesson 2 of 8

## Big Idea: This is a fast-paced lesson that meets students where they are and challenges them to cement their knowledge of solving and creating linear equations.

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Standards:
Subject(s):
Math, linear inequalities, Algebra, linear equation, problem solving, assessment, group work, word problems, conceptual approaches
43 minutes

### James Dunseith

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