# Middle Grade examples

The following examples were taken from the Illustrative Mathematics website. They were picked out to try and form a basis for discussion and to give people some ideas on what they can be doing with CCSS between meetings. These aren’t all the problems (or necessarily the “best”) but they do all have a context to help kick-start some discussion.

Sixth Grade:

- For continued work with with ratios and fractions this problem has several variations (2, 3, and 4) as well as solutions that suggest several different ways of explaining the problem to students with varying levels of comfort with fractions. (Also here is a separate example that could be another way of explaining this type of problem).
- A Ratios problem that doesn’t involve fractions (and could be used to get a discussion going among the class) can be found here.
- An interesting division problem may be found here. This problem has students look into what the division algorithm
*means*an important bit of “juice” that will help students develop a mastery of not just division, but algebra in general. And here are a couple related examples: link 1 and link 2. - Links to a progression set on 6th grade Geometry are here: 1, 2, 3, and 4. This set serves to highlight how students should be encouraged to think about problems in terms of what they have already done.

Seventh Grade:

- Here are some 7th grade fraction problems: A potentially complicated question, a more straight-forward question with a variety of solutions, and a flexible example that can be tweaked to be more of a unit conversion problem, if desired.
- This problem is a good example of helping students visualize and conceptualize algebra as well as to get more familiar with the number system. As is fairly typical for Illustrative Math problems, a little adjustment could make this a more complicated problem later in the term. Some suggestions would be adding a third variable, having the labeled points be fractions, negative, or just not zero and/or one, or asking the student to compare different quantities instead of just labeling each as negative or non-negative. For example: “which is greater a-b or b-a ?” can build upon knowledge about which of those is positive and which is negative, but isn’t directly asking students to think about that. This begins to cross into the realm of Expressions and Equations problems such as this or this.
- A sometimes-rare algebra problem involving an
*inequality*can be found here. - This example is a Geometry problem that uses Rates and Proportions to be a little more complex. It has a variety of solutions, which is always helpful for students who may be struggling with some aspects of the problem. This problem doesn’t have a variety of solutions, but it does require some thought and relation to “other” areas of math would definitely help students work towards a solution.

Eighth Grade:

- As a start to functions this problem has two different ways of conveying the same required information to students and has strong connections with Algebraic Expressions and Equalities that can help show students a different way of going about the same (or similar) problems. This example takes another approach to thinking about functions. There would also be potential to add some graphing as yet another way to have students think about that problem (such as the story aspect of this problem).
- Some 8th grade fractions and unit conversions are here. While this Expressions and Equations (read: algebra) problem could provide some discussion of what “reasonably close” really is. Having students justify their yes/no answer might seem like an exercise for English class (to them!), but reinforcing that explanation can be the difference between right and wrong is valuable. Along a similar line of thinking (pun intended) this problem could be given a similar context to the burger problem in order to continue discussions of things like “close” and “equal” that can, in real-world examples, be a bit fuzzy. (Related example that could also start discussions of non-linear equations, if the class is ready for it).
- The Pythagorean Theorem (and this problem associated with it) can be used to tie together a lot of the math students do in the 8th grade. Graphing, “reasonably close,” and functions can all be made to involve the Pythagorean Theorem.

Further examples may be added as they come to the attention of administrators, but examples generated by this group of teachers and UO-affliates will be found below!

**Division of Fractions** or **“The Watermelon Problem”** which was created by Kristy McElravy and Ryan Roulston:

In order to help students gain a better conceptual understanding of what a division problem such as ½÷¼ really *means* this problem explains the situation in terms of dividing half of a watermelon into bowls that will hold only a quarter of a watermelon.

Once students become familiar with answers that would be interpreted as a whole number of bowls required to hold some amount of watermelon this example can be modified to introduce more complex relationships. For example, students can be asked to note how many bowls, and how full the last bowl would be if he/she had 3/4 of a watermelon and bowls that held 1/2 of a watermelon.

Other phrasings of this problem shift the focus to different ways in which fractions may be interpreted. Some of these phrasings address concepts that come earlier or later in CCSS.

- 5 watermelons being put into bowls which hold 3/8 of a watermelon (whole # ÷ fraction)
- 1/2 of a watermelon being spread equally into 4 bowls, how much of a watermelon should each bowl have? (fraction ÷ whole #)
- Can 12/8 of a watermelon be equally divided into bowls that hold 1/4 of a watermelon so that all the bowls used are full? (conceptual question that could be phrased more blandly as: does 1/4 divide 12/8 evenly?)

As a longer, possibly more advanced question about fractions that could have the numbers tweaked to change the true/false answer part:

- If Sally has 12/8 of a watermelon and 3 bowls that can only hold up to 1/2 of a watermelon can she fit her watermelon in her bowls? If she fills them all equally, how much of a watermelon would each bowl have?

As a few ways to continue this problem and make students think:

- If Sally finds a 4th bowl she forgot about, can she fit the watermelon in her bowls? If she fills them all equally, how much would each bowl have now?
- If Sally gets hungry and eats 1/5 of a watermelon while cutting it up for her bowls… How much does she have left? Will the watermelon fit in her bowls now? If she equally divided the rest of the watermelon, how full would each bowl be?

And that concludes this example of division of fractions!