# Home Towns in Texas

Amy, Clara, Joel, Eric, Ryan, and Brody are students attending a summer math camp together. Each student comes from a different town in Texas. The students work in teams of two to determine which team comes from towns with the greatest combined population. The teams also decide to use the greater than and less than symbols to compare their home town populations. The students look online to find the population of each of their home towns. Here is what the students find out.

__Team 1__

Amy lives in Odessa which has a population of 99,940.

Clara lives in Denton which has a population of 113,383.

__Team 2__

Joel lives in Richardson which has a population of 99,203.

Eric lives in College Station which has a population of 93,857.

__Team 3__

Ryan lives in Amarillo which has a population of 190,695.

Brody lives in Grapevine which has a population of 46,334.

The students decide to round each population number to the nearest hundreds place to make calculating easier. Which team lives in towns with the greatest combined rounded population?

Team 1 uses the greater than or less than symbol to compare the exact populations of their two towns. What statement does Team 1 write?

Team 2 uses the greater than or less than symbol to compare the rounded populations of their two towns. What statement does Team 2 write?

Team 3 uses the greater than or less than symbol to compare the exact and rounded populations of their two towns. What statements does Team 3 write?

Show all your mathematical thinking.

## Plan

#### Underlying Mathematical Concepts

#### Possible Problem-Solving Strategies

#### Formal Mathematical Language and Symbolic Notation

### Possible Solutions

## Solutions tabs

### Assess

### Anchor Papers

#### Novice 1

### Scoring Rationale

#### Problem Solving

#### Novice

The student's strategy of using table to indicate the population stated in the task for each student and replacing the ones, tens, and hundreds places with zeros in the "round number" column does not work to solve the first part of the task. The student's answer, "Amy has the most population because she has 3 nines __99,9__40 on my table," is not correct. The student does not provide a strategy to solve the second part of the task. The student's statement, "Amy is > everyone," is not correct.

#### Reasoning & Proof

#### Novice

The student is able create a table to list the six students and scribe the correct population from the task but demonstrates no understanding of the underlying concepts of the task. The student does not demonstrate understanding that the six students form three teams. The student does not demonstrate understanding of the mathematical concept of rounding to the hundreds place. The student replaces all the ones, tens, and hundreds place value numbers with zeros. The student is not comparing numbers by place value but by the magnitude of the first numbers reading from left to right. Therefore, 99,940 is considered the greatest population. The student does not demonstrate understanding of the concept of using greater than and less than to compare the exact and rounded town populations between two students.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms *population, greater than, symbol* from the task. The student also correctly uses the term *table*. The student does not use the mathematical symbol, >, correctly.

#### Connections

#### Novice

The student solves the task and does not attempt to make a mathematically relevant observation about her/his solution.

#### Representation

#### Apprentice

The student's table is appropriate to part of the task but is not accurate. The third column should be a labeled rounded number of population. All the entered data for the third column is not accurate.

#### Overall Achievement Level:

#### Novice

#### Apprentice 1

### Scoring Rationale

#### Problem Solving

#### Apprentice

The student's strategy of using a table to indicate the exact and rounded populations of six towns and applying addition to determine the team with the greatest combined population works to solve the first part of the task. The student's answer, "Team 3 has a greater population," is correct. The student's strategy of using a table to compare the exact and rounded populations using greater and less than symbols per student would not work to solve the second part of the task. The student's answer, "I compared every town with the > and < symbols," is not correct.

#### Reasoning & Proof

#### Apprentice

The student demonstrates understanding of the first part of the task by correctly applying the concept of rounding to the nearest hundred for populations of five towns. The student's error for College Station is considered a careless error and not a flaw in the student's reasoning. The student uses addition to calculate each team's total rounded population and determines the team with the greatest combined population. The notation flaw for College Station does not lead to an incorrect answer to the question. The student does not show correct reasoning in the second part of the task. The questions require the student to compare the exact population, the rounded population, or the exact and rounded population between each teams' two home towns. This student is just comparing the exact and rounded populations per student.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms *population, greater/less than, symbols, number, greatest* from the task. The student also correctly uses the terms *table, least*. The student correctly uses the mathematical symbols, < and >.

#### Connections

#### Practitioner

The student makes mathematically relevant observations about her/his solution. The student states, "I noticest that Ryan lives in Amarillo and it has the greatest number of people living there," and "His team friend lives in Grapevine and has the least people living there."

#### Representation

#### Apprentice

The student's first table is appropriate to the task but is not accurate. The student provides all necessary labels but the entered data for the rounded population for Eric from College Station is not correct. It should state 93,900. The student's second table is not appropriate to the task and has errors. The fourth column should be labeled rounded population. Eric's rounded population should read 93,900. Ryan's population should read 190,695 and his rounded population should read 190,700.

#### Overall Achievement Level:

#### Apprentice

#### Practitioner 1

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of using a table to indicate the exact and rounded populations of six towns, applying addition and the correct use of the greater and less than symbols works to solve this task. The student's answers, "Team 3 has more population than Teams 1 and 2," "Team 1 Amy 99,940 < 113,383 Clara," "Team 2 Joel 99,200 > 93,900 Eric," and "Ryan 190,695 > 46,334, Brody, Ryan 190,700 > 46,300 Brody," are correct.

#### Reasoning & Proof

#### Practitioner

The student demonstrates correct understanding of the first part of the task by applying the concept of rounding to the nearest hundred to the population of six towns. The student uses addition correctly to calculate each team's total rounded population and determines the team with the greatest rounded combined population. The student shows correct reasoning of the second part of the task by comparing the exact and/or rounded town populations for each team and using the greater than and less than symbols correctly in stating the comparisons.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms *greatest, greater than, less than, symbols, population* from the task. The student also correctly uses the terms *total, table, more than, most, least, 10,000s place*. The student correctly uses the mathematical symbols, < and >.

#### Connections

#### Practitioner

The student makes mathematically relevant observations about her/his solution. The student states, "Ryan's town has the most population," "Joel's has the least population," and "Amy Joel Eric and Ryan have a 9 in the 10,000s place."

#### Representation

#### Practitioner

The student's use of a table is appropriate to the task and accurate. The student provides all necessary labels and the entered data is correct.

#### Overall Achievement Level:

#### Practitioner

#### Practitioner 2

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of using a table to indicate the exact and rounded populations of six towns and applying addition works to solve the first part of the task. The student's strategy of using number lines and the greater than and less than symbols works to solve the second part of the task. The student's answers, "Ryan and Brody have the most population rounded total. They are team 3" "99,940 < 113,383," "99,200 > 93,900," "46,334 < 190,695," and "46,300 < 190,700," are correct.

#### Reasoning & Proof

#### Practitioner

The student demonstrates correct understanding of the first part of the task by applying the concept of rounding to the nearest hundred for the population of six towns. The student uses addition correctly to calculate each team's total rounded population and determines the team with the greatest rounded combined population. The student shows correct reasoning of the second part of the task by comparing the exact and/or rounded town populations for each team and using the greater than and less than symbols correctly in stating the comparisons.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms *population, greatest, symbols* from the task. The student also correctly uses the terms *most, total, number lines, table, 100 thousands place, place value, least*. The student correctly uses the mathematical symbols, < and >.

#### Connections

#### Practitioner

The student makes mathematically relevant observations about her/his solution. The student states, "You don’t have to add team 2 because they have no 100 tousands place and I can see they can't go past 1 in the 100 thousands total," "Ryan's town has greatest population," "I counted by 20,000 on team 3 number lines," and "Brody has least population."

#### Representation

#### Practitioner

The student's use of a table is appropriate to the task and accurate. The student provides all necessary labels and the entered data is correct. The student's four number lines are appropriate to the task and accurate. Each number line has the team labeled, numbers labeled as either exact population or rounded population, and the intervals on the number lines are accurate and labeled correctly.

#### Overall Achievement Level:

#### Practitioner

#### Expert 1

### Scoring Rationale

#### Problem Solving

#### Expert

The student's strategy of using a table to indicate the exact and rounded populations of six towns and applying addition works to solve the first part of the task. The student's strategy of using tables and the greater than and less than symbols works to solve the second part of the task. The student's answers, "team 3 has the most population sum," "99,940 < 113,383," "99,200 > 93,900," "190,695 > 46.334," and "190,700 > 46,300," are correct. The student brings prior knowledge of data and fractions to her/his solution.

#### Reasoning & Proof

#### Expert

The student demonstrates correct understanding of the first part of the task by applying the concept of rounding to the nearest hundred for populations of six towns. The student uses addition to calculate each team's total rounded population and correctly selects the team with the greatest population. The student shows correct reasoning for the second part of the task by comparing the exact and/or rounded town populations for each team and using the greater than and less than symbols in stating the comparisons. The student takes time to justify that her/his rounding was correct by discussing how one must be sure to be looking at the same place value when comparing two numbers. The student brings data concepts to her/his solution by including range, minimum, and maximum. The student also uses the fraction concept of one-half correctly.

#### Communication

#### Expert

The student correctly uses the mathematical terms *population, greatest, symbols* from the task. The student also correctly uses the terms *tables, most, sum, difference, data, tens place, number, range, minimum, maximum, thousand, third*. The student correctly uses the mathematical symbols, <, >, 1/2.

#### Connections

#### Expert

The student makes mathematically relevant Practitioner observations about her/his solution. The student states, "Joel's population was a difference by only 3 people so it is really the most acerite [accurate] in real life. The student computes the exact and rounded population difference between Odessa and Richardson and states, "Closest population of towns." The student also states, "rounding to the tens place would make populations closer for each town in exact and rounding numbers." The student makes Expert connections. A warning is provided by the student to inform that one should be careful to compare the same place value positions when comparing numbers. It appears the student follows her/his advice as she/he states, "I almost picked Odessa so want to warn other third graders." The student includes data and fraction concepts in her/his solution.The student states, "That is importent when you do data," when discussing the difference in Joel’s exact and rounded populations. The student also states, "When you round down you miss people data. Eric rounded up so his town has 43 people added to the real population." The student continues by stating, "the population range is from 46,334 (Grapevine, minimum) to 190,695 (Amarillo, maximum)." The student includes fractions in her/his solution by stating, "Grapevine population is about 1/2 of College Station."

#### Representation

#### Expert

The student's use of a table to show the exact and rounded populations for all six students is appropriate to the task and accurate. The student provides all necessary labels and the entered data is correct. The student's four additional tables are appropriate and accurate. The student uses her/his "big table" to help extend thinking to data and fractions.