Problems to Ponder

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  • 1 Lincoln Public Schools in Lincolns, Nebraska
  • 2 Denver Public Schools in Denver, Colorado
  • 3 Denver Public Schools

Problems to ponder

1. Look through your favorite picture book. What rectangles do you see? Tell your friends how you know each one is a rectangle.

2. Draw a picture of your family. How many eyes are there?

3. Mateo saw these two shapes:

His friend said the shapes are the same. Use your words to tell why you think his friend is correct or incorrect.

4. Brett got five goldfish for his fifth birthday. He has two small tanks. How many goldfish could Brett put in each tank? Is there more than one way?

5. Kalli covers three numbers on a 100 chart. All of the three numbers she covers are greater than 73. What numbers might be covered? Use pictures, numbers, or words to tell or show how you know.

6. A true equation is: 3 + 4 = 7

A false equation is: 6 = 2 + 1

Can you write four true equations using the numbers 2, 4, 6, and 8?

7. Raj is solving 84 – 37. He gives 3 to 84 to make 87. Next, he subtracts 37 from 87 to get 50. Then he gives back the 3 to get 47. Does Raj's strategy work? Why or why not?

Can you think of another strategy to use to solve this problem?

8. Draw a number line. On one end, mark 200, and on the other end, mark 400. About where would 275 be on the number line? 368? 223? Show where you would place each number and share your reasoning.

9. Kipton goes outside to play basketball at 4:15 p.m. He plays for 1 1/2 hours. Then he plays baseball for 15 minutes before going inside. What time did Kipton go inside? (You may want to use an open number line to solve this problem.)

10. Robert Wadlow holds the record for having the biggest feet. He wore a size 37 shoe! His left foot was 18 1/2 inches long. Measure yovur left foot. How much shorter is your left foot than Robert's left foot? Give your answer to the nearest 1/4 inch.

11. Bakin is drawing a design with shapes. His only direction is that he has to use shapes that do not have more than 10 sides and each shape has to have least one set of parallel lines. What shapes might Bakin use to create his design?

12. Think about the mixed number 5 3/5. How many fractions can you think of that name the same amount?

13. A company that manufactures boxes for cereal wants to come up with a new design. They have the following options for dimensions for the new boxes:


Which dimensions would you recommend to the company if they want to make a cereal box with the largest volume? What if they wanted a cereal box with the smallest volume? Be ready to explain how you found your answers. (Using unit cubes may be helpful to solve this problem.)

14. Lindsay drew a rectangle. She drew another rectangle that was double the length and double the width of the first rectangle. She noticed something interesting about the area of the new rectangle, so she drew a third one. For this third rectangle, she doubled the length and width of the second rectangle. She repeated this process to draw a fourth rectangle. What do you think Lindsay was noticing about the relationship between the areas of the rectangles she made? Why might this be happening?

15. A barge loads trailers from semitrucks for overseas shipments. The barge can carry trailers stacked 10 deep, 9 wide, and 3 tall. The crane loads one trailer at a time and takes 43 seconds to complete this task for each trailer. If the barge begins loading trailers at 3:32 p.m., what time does it finish loading?

16. Maria is shopping for a new dress at her favorite store. She finds a dress that was originally priced at $36.99 but has been marked down 20 percent. Today, all sale items are discounted an additional 15 percent off the sale price. What is the price of the dress today?

17. Alex claims that to find half of one-third of a candy bar broken into six congruent pieces he first found half of the bar (3 pieces) and then one-third of the bar (2 pieces). Alex then subtracted the values to get one of six pieces or one-sixth of the candy bar. Does this process always work? If not, for which cases does it work?

18. Each side of a 3-4-5 right triangle forms the diameter of a circle. What is the ratio of the areas of the circles?

19. A kite is twice as tall as it is wide. If the wingtips are located m units away from the top of the kite, vertically, and m is 1/3 of the height of the kite, what expression models the difference in side lengths of the kite?

20. The point A(12, 5) is reflected across the x-axis, and then its image is reflected across the y-axis. What is the length of the line connecting point A and point A

21. Five years ago, Alicia was twice as old as her brother Tommy. If Alicia is 13 years old, how old is Tommy?

22. Given 2x + a = b and x + a = 8, what do we know about the relative values of a and b?

23. A medium-size dog is tethered to the middle of his doghouse opening, which is centered on the short side of the doghouse. The base of the doghouse is a rectangle measuring 30 inches by 35 inches. If the dog's leash is 5 feet long and tethered to the house neck-high, approximately how many square feet does the dog have access to?

24. Each side of a 45-45-90-degree right triangle forms the diameter of a circle. How many times larger is the area of the largest circle than the smallest circle?

25. Elizabeth's yard is square. She has a sprinkler head in each of the four corners, and the radius of the spray from the sprinkler head is the length of her yard. However, one of the heads is broken. Elizabeth decided to turn off the head in the opposite corner and use two sprinklers to water the yard. What expression represents the area of overlap?

26. A line with a positive slope intersects the parabola y = –(x – 1)2 + 2 at (–1, –2) and (m, n) such that m and n are whole numbers. How many points (m, n) could define the intersection of the line and the parabola?

27. A pencil is placed at center of a circle with radius n, tangent to both the x- and y-axes. If the circle is rotated about the origin 360 degrees, what is the distance the pencil has traveled?

28. For which values of x and y is ln(xy) = (x/y)?

Susie Katt,, works as a K–2 mathematics coordinator for Lincoln Public Schools in Lincoln, Nebraska.

Megan Korponic is the High School Mathematics Curriculum Specialist for Denver Public Schools in Denver, Colorado.

Cathy Martin,, serves as the Executive Director of Curriculum and Instruction for Denver Public Schools.

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Supplementary Materials

Mathematics Teacher: Learning and Teaching PK-12


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