What are the Magic Words?To findTo getTo figure outTo showBecause Since Therefore

Sample 5th Grade Question

• Some 5th graders competed in a field-day race. The 5th grade results for the 50-meter race are shown in the chart.(Standards 2.1, 2.5, 2.6)

Sample 5th Grade Item

8.72Donna8.28Matt

7.86Rose

8.98José

7.68Tia

8.05Andrew

Time in SecondsRunner

Sample 5th Grade Item• List the runners from fastest to slowest.

REMEMBER: the runner completing the race in the least amount of time is the winner.

• CALCULATE the average time for the 5th grade. Which runner’s time was closest to the average for 5th grade? CALCULATE by how much time this student’s time differed from the average.

The Directions• For full credit, you must do the following:

1. Show OR describe each step of your work, even if you did it in your head (“mental math”) or used a calculator

2. Write an explanation stating the mathematical reason(s) why you chose each of your steps

Solving Open-ended Items

Tips from Teachers

Solving Tips from Teachers• Show and number each step of the work

– Even/especially if the work was done in the student’s head or calculator

Tips for Solving• Make 2 columns

– Put “work” on the left– Put “explanation” on the right

work explanation

The Work (a student’s response)

A) 1) Fastest 1. Tia 7.68 sec

2. Rose 7.86 sec3. Andrew 8.05 sec4. Matt 8.28 sec5. Donna 8.72 sec

Slowest 6. José 8.98 sec

The Work (a student’s response)

2) 7.68 + 7.86 + 8.05 + 8.28 + 8.72 + 8.98 = 49.57 sec

3) 49.57 ¸ 6 = 8.26 sec (average)

4) Matt’s time 8.28 sec is the closest

5) 8.28 – 8.26 = .02 sec

Explanation Tips

• Make sure ALL steps are explained in words

• Encourage your students NOT to use the words “to find the answer”– The words “the answer” do not explain what the

answer represents

Explanation Tips

• Encourage students to EXPLAIN their work - not DESCRIBE it– Description:

“I divided 49.57 by six which equals 8.26”– Explanation

“I divided the total time by the number of runners TO FIND the average time”

Explanation Tips

• Explain which operation was performed and what each number means

– Start each step with “I added…”“I subtracted…”“I multiplied…”, etc

– Then explain what each number represents“I divided the total time by the number of runners…

Explanation Tips

• Use “magic words”* in the explanation

• These are words that gear students to ‘explain’ their work rather than ‘describe’ it

* Article on “Magic Words” can be found in the math assessment handbook

What are the Magic Words?

• To find• To get• To see• To figure out• To show• To determine• Because • Since • Therefore…….

The Explanation(a student’s response)

• I ordered the runners from fastest to slowest. Tia won the race. The runners up in order were Rose, Andrew, Matt, Donna and last was José.

• I added all the runners’ times together TO FIND the total time.

The Explanation (a student’s response)

3) I divided the total time by the number of runners TO GET the average time for 5th grade.

4) I compared the runner’s time to the average and I see that Matt’s time was the closest to the average.

5) I subtracted the average time from Matt’s time TO

FIGURE OUT how much he differed from the average.

The Final Product

work

A) 1) Fastest 1. Tia 7.68 sec 2. Rose 7.86 sec 3. Andrew 8.05 sec 4. Matt 8.28 sec 5. Donna 8.72 sec

Slowest 6. José 8.98 sec

B) 2) 7.68 + 7.86 + 8.05 + 8.28 + 8.72 +8.98 = 49.57 sec

3) 49.57 ¸ 6 = 8.26 sec (average)

4) Matt’s time 8.28 sec is the closest

5) 8.28 – 8.26 = .02 sec

explanation

1) I ordered the runners from fastest to slowest. Tia won the race. The runners up in order were Rose, Andrew, Matt, Donna and last was José.

2) I added all the runners times together TO FIND the total time.

3) I divided the total time by the number of runners TO GET the average time for 5th grade.

4) I compared the runner’s time to the average and I see that Matt’s time was the closest to the average.

5) I subtracted the average time from Matt’s

time TO FIGURE OUT how much he differed from the average time.

Open-ended Items

In Your Math Classroom

Where to Find Open-Ended Problems

• Mathematics Assessment handbook & Released Items Handbook– www.pde.state.pa.us– Click on “K-12 Schools”– Click on “Assessment & Testing”– Click on “Assessment Handbooks”

• Mathematics Instructional Rubrics (Primary thru High School)– www.pde.state.pa.us– Click on “K-12 Schools”– Click on “Curriculum & Instruction”– Click on “Mathematics”– Click on “What’s New”

Where to Find Open-Ended Problems

• Open-ended problems created at the 2002 Math Governor’s Institutes– www.pde.state.pa.us– Click on “K-12 Schools”– Click on “Curriculum & Instruction”– Click on “Mathematics”– Click on “Professional Education”

• PSSA Practice Tests (5th, 8th & 11th)

Where to Find Open-Ended Items

• Text resource books• Multi-step word problems from your textbook• Teacher-made questions• Internet resources

The Importance of

Mathematics Vocabulary

• Mr. Stevens bought 15 books for his store. He paid $11 for each book. He then sold all 15 books for $25 each.

• A. What is the difference between the total amount of money Mr. Stevens sold the books for and the total amount of money he paid for the books? Show or explain all your work.

• Part A Answer Support• $210• $11 × 15 = $165• $25 × 15 = $375• $375 – $165 = $210• OR• ($25 – $11) × 15 = $210 • OR• He paid $11 times 15 ($165) and sold the books for $25 • times 15 ($375). I subtracted $375 from $165 to get $210 • as the difference.

Sample QuestionFour members of the Johnson family took a trip from Pittsburgh to Harrisburg, a distance of 221 miles. It took them 4 hours and 15 minutes to make the trip. The car required 13 gallons of gasoline at $1.25 per gallon. The turnpike toll was $6.50, and they spent $12.84 for food. What was the average cost per mile based on the total expenses of gas, food and tolls for this trip?

The Work1) $1.25 X 13 gal = $16.252) $16.25 + $6.50 + $12.84 = $35.593) $35.59 221 mi $0.16104074) $0.16 per mile

Explanation TipsEncourage students to EXPLAIN their work -not DESCRIBE it• Description:“I multiplied $1.25 and 13 and got $16.25”• Explanation“I multiplied the price of gas per gallon and the number of gallons to get the price for the gas used.”

The Explanation1) I multiplied the price of gas and the number of gallons TO GET the total cost of gas.2) I added the cost of gas, food and tolls together TO FIND the total cost of the trip.3) I divided the total cost of the trip by the number of miles and I FOUND the cost per mile.4) SINCE I had many decimal places, I rounded to the hundredth BECAUSE the answer was in terms of money. My answer is 16 cents per mile.

The Final ProductWork1) $1.25 X 13 gal = $16.252) $16.25 + $12.84 + $6.50= $35.593) $35.59 221mi $0.1610414) $0.16 per mile

Explanation1) I multiplied the price of gas and the number of gallons TO GET the total cost of gas.2) I added the cost of gas, food and tolls together TO FIND the total cost of the trip.3) I divided the total cost of the trip by the number of miles and I FOUND the cost per mile.4) SINCE I had many decimal places, I rounded to the hundredth BECAUSE I wanted money. My answer is 16 cents per mile.