Big Ideas

Cheryl wrote,

Do you know of any good activities that are appropriate for 6th graders that help explain/show why you divide the numerator by the denominator to convert a fraction to a decimal?

Cheryl,

What a great question. And I’ll answer in the most roundabout of ways. We get numbers from counting, measuring and performing operations on those numbers.

• Counting usually gives us whole numbers
• Measuring with the metric system or money gives us decimals
• Measuring with the standard system gives us fractions
• For operations like division, we can choose—when we divide 7 by 2 we can stick with whole numbers 3 with remainder 1, use fractions 3 1/2, or use decimals 3.5—it just matters what you want your units to be wholes, halves, or tenths. BUT, and here’s the segue to your question—they are all just equivalent representations for the same number.

So, if I think of fractions, decimals, and percents as equivalent representations, then I get use my favorite equivalent picture to think about this idea. Have you ever met the ratio table?

There are strategies to getting from one column to the next, * or ÷ by a number or + or – two columns. In this case, I am using the * or ÷ by a number strategy.

If I have a 1 as the whole, the part is represented in decimal form
If I have a 100 as the whole, the part is represented in percent form
If I have any other number as the whole, then we get a fraction form.

I could share a podcast with you or a reference to more ratio table materials and applets if you aren’t familiar with this one.

Share your thoughts if you try any of it….

Cheryl asks,

“We just finished teaching adding and subtracting integers to the 7th graders and we had a quiz today.  I got wonderful explanations from the kids.  For example, one of my students said that -19 – (-34) = 15 because: “You are short 15 negatives, so you add  15 negatives.  To keep the same net value, you add 15 positives too.  When you finish -34 – (-34) = 0.  You have 15 positives left over, so the answer is positive 15.

We are moving onto multiplication and division on Monday and the other teacher and I want to use the concept of multiplication as grouping as the jumping off point.  This model seems to break down, though, when you have a problem like 6 / -2.  If you read it as 6 things broken into the opposite of two groups it can work.  The two groups would have 3 positives each and the opposite of this would be 3 negatives per group.  However, we cannot figure out a way to describe it if it is read as 6 things broken into the opposite of groups of 2.  There is no such thing as the opposite of groups of 2 — whether the things are positive or negative there are still 2 things per group.  Apparently, last year they got around this fact by having the students rewrite division problems as multiplication — 6 / -2 became ____ x -2 = 6.  However, we would like to be consistent in our explanations if we can.”

How great that you are trying to make sense of all of this in some way or another. I struggle with this myself, surprisingly often, but your wording has given me an idea. You wrote, “ If you read it as 6 things broken into the opposite of two groups it can work.”

What if you tweak that to say, “6 things broken into two groups of opposites?”

So if you break 6 into 2 groups, then you get 3 in each group, but it’s of opposites, so you get 3 opposites or -3?

Try it out a couple of times and tell me how it feels. I have never used this terminology before, so no promises on my part.

Kim asks,

We are studying inequalities and I need to explain to my students WHY, when we multiply or divide an inequality by a negative, that we flip the inequality sign?

Imagine a number line on the board. Put your thumb on the 2 and your pointer on the 4.

Now, add 3 to both of these. Can you imagine your fingers sliding to 5 and 7—sliding together not changing their relationship to each other?

Now, subtract 1 from both (or add -1). Can you imagine your fingers sliding to 4 and 6—sliding together, not changing their relationship to each other? The pointer is still farthest to the right and they are the same distance apart.

Now multiply both of these by 2. Can you imagine sliding your fingers to 8 and 12—sliding them both in the same direction, but altering their relationship to each other. The pointer is still farthest to the right but the distance between them got bigger.

Now multiply both of them by 1/4 (or divide by 4). Can you imagine sliding your fingers to 2 and 3—sliding them both in the same direction, but altering their relationship to each other? The pointer is still farthest to the right but the distance between them gott smaller.

Now multiply both of them by -1. Can you imagine sliding your fingers to -3 and -2—sliding them both in the same direction, but altering their relationship to each other? The distance between them stayed constant, but the order of them changed.

You could then try combinations of these, like multiplying by -1/2 or -2.

What do you guys think? Anyone willing to try it and report back?