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# Thousands and Millions of Fourth Graders

Alignments to Content Standards: 4.NBT.A.1 4.OA.A.1 4.NBT.B.5

There are almost 40 thousand fourth graders in Mississippi and almost 400 thousand fourth graders in Texas. There are almost 4 million fourth graders in the United States.

We write 4 million as 4,000,000. How many times more fourth graders are there in Texas than in Mississippi? How many times more fourth graders are there in the United States than in Texas?  Use the approximate populations listed above to solve.

There are about 4 thousand fourth graders in Washington, D.C. How many times more fourth graders are there in the United States than in Washington, D.C.?

## IM Commentary

The purpose of this task is to help students understand the multiplicative relationship between commonly used large numbers (thousands and millions) by using their understanding of place value. This task also connects to students' work on multiplicative comparison (see 4.OA.A.1). The task 4.NBT Threatened and Endangered is a good task to do before this one as it requires the same kind of reasoning but the numbers are smaller. The numbers in this task come from the National Center for Educational Statistics:

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This particular task supports the demonstration of Mathematical Practice Standard 6, Attend to precision.  Fourth graders apply their understanding of place value and multiplicative reasoning to compare thousands and millions in this task. They need to pay close attention to precision to determine the value of each digit in each number and to compare the two numbers. They make use of the fact that the value of each place is ten times the value of the place immediately to the right and are able to explain this using precise language.

## Solution

​We write 4 thousand as 4,000

We write 40 thousand as 40,000

We write 400 thousand as 400,000

The value of each place is ten times the value of the place immediately to the right.

So:

40,000 is 10 times 4,000

400,000 is 10 times 40,000.

4,000,000 is 10 times 400,000.

Thus, 400,000 = 10 $\times$ 40,000, and there are about 10 times as many fourth graders in Texas as there are in Mississippi.

Also, 4,000,000 = 10 $\times$ 400,000, and there are about 10 times as many fourth graders in the US as there are in Texas.

Finally, to go from 4,000 to 4,000,000, we have to multiply by 10 three times. We see that $$10\times10\times10=10\times100=1000$$ So there are about 1,000 times as many fourth graders in the US as there are in Washington DC.

#### Jayesh says:

over 3 years

Also, the progressions stress that "x is 1,000 times bigger than y" statements are ambiguous, since though it is understood as meaning the value of x is 1,000 times the value of y, it technically means the difference of x and y is 1,000 times the value of y. Should "bigger than" statements here be modified to "as big as" statements? Otherwise, a note in the commentary explaining that this task is intended for instruction rather than assessment might be appropriate, in order to justify the less formal language. Hi redbaron,

I tried to find reasonable wording that is consistent with the recommendations of the progression, but was not successful. Perhaps you have specific wording suggestions? The progression notes that acceptable language is changing, and I suspect the reason for this is that there is no way to word these questions that is consistent with the recommendations without sounding pedantic. If you can offer a concrete suggestion for wording, that would be great. (I personally don't find the language ambiguous.) My suggestion was to use "as... as" phrases. In the first sentence in the task, this would be "How many times as many fourth graders are there in Texas as in Mississippi?" I think wordings such as this could come across as careful, but not so much that would get in the way of understandability, or appear that we are trying too hard.

The progressions indeed do make a concession about "times more than" phrases becoming acceptable in printed publications, but recommend more cautious language in testing situations even so. This is why I wondered if it would be best to simply address the informal language in the commentary, stating that it favors instructional purposes. Since the standards for grade 4 explicitly limit the generalization of place value to numbers less than or equal to 1,000,000, I wonder if this should either be in a later grade or at least tied to the cluster level? If not, a description in the commentary about the alignment choice would be helpful. Thanks, Director of Mathematics, Smarter Balanced, for noting this. I changed the task to fit within the scope of 4th grade work, and copied the old version into a 5th grade task here:

https://www.illustrativemathematics.org/illustrations/1931

Let us know if you see any other issues.

Me to

#### redbaron says:

over 4 years

Also, the progressions stress that "x is 1,000 times bigger than y" statements are ambiguous, since though it is understood as meaning the value of x is 1,000 times the value of y, it technically means the difference of x and y is 1,000 times the value of y. Should "bigger than" statements here be modified to "as big as" statements? Otherwise, a note in the commentary explaining that this task is intended for instruction rather than assessment might be appropriate, in order to justify the less formal language.

#### Kristin says:

over 4 years

Hi redbaron,

I tried to find reasonable wording that is consistent with the recommendations of the progression, but was not successful. Perhaps you have specific wording suggestions? The progression notes that acceptable language is changing, and I suspect the reason for this is that there is no way to word these questions that is consistent with the recommendations without sounding pedantic. If you can offer a concrete suggestion for wording, that would be great. (I personally don't find the language ambiguous.)

#### redbaron says:

My suggestion was to use "as... as" phrases. In the first sentence in the task, this would be "How many times as many fourth graders are there in Texas as in Mississippi?" I think wordings such as this could come across as careful, but not so much that would get in the way of understandability, or appear that we are trying too hard.

The progressions indeed do make a concession about "times more than" phrases becoming acceptable in printed publications, but recommend more cautious language in testing situations even so. This is why I wondered if it would be best to simply address the informal language in the commentary, stating that it favors instructional purposes.

#### Shelbi says:

over 4 years

Since the standards for grade 4 explicitly limit the generalization of place value to numbers less than or equal to 1,000,000, I wonder if this should either be in a later grade or at least tied to the cluster level? If not, a description in the commentary about the alignment choice would be helpful.

#### Kristin says:

over 4 years

Thanks, Director of Mathematics, Smarter Balanced, for noting this. I changed the task to fit within the scope of 4th grade work, and copied the old version into a 5th grade task here:

https://www.illustrativemathematics.org/illustrations/1931

Let us know if you see any other issues.

over 3 years

Me to