Problems to Ponder

These are the March 2020 P2P problems from Steve Ingrassia and Molly Rawding.

1 Find a clock with hands. Count up to 12. Then count backward from 12.

2 Here is a pattern with colors and shapes.

What repeats?

Draw a pattern with two different colors. How many steps are in your pattern before it repeats?

3 Draw three animals. What can you count? Legs? Eyes? What else?

4 Take 10 counters. Put the counters in two piles. How many counters are in each pile? Try this another way. Make drawings of all the possible ways.

For further thought: What if you make three piles? How many counters can you put in each pile? Is it still 10 altogether?

5 Hannah was going on a walk through the woods. When she got home, she told her brother that she saw 20 legs. If the animals that Hannah saw included ducks (2 legs), chipmunks (4 legs), and spiders (8 legs), what might Hannah have seen on her walk?

6 Which design costs the most?

For further thought: What designs can you make that are worth 13?

7 Andrea has some money. Her brother gives her 7 coins worth 23 cents and now she has 73 cents. If she started with 5 coins, what coins did Andrea's brother give her and what coins did Andrea start with?

8 From one end of the street, Marian's house is the ninth house on the right. From the other end of the street, it's the fifth house on the left. Draw a picture of Marian's street. How many houses are on Marian's side of the street?

9 In a pattern of numbers decreasing in value, 48 is the third number.

Complete the pattern in two different ways.

__ _,_ __,48,__ _,_ _____,___,48,___,__ _

10 Imagine there is a new operation >

5>6=16 6>4=1610>1= 211>10=21

Find the value of each question mark:


For further thought: What could be the values of g and m?


11 Laurel is planning a family reunion. She has tables that seat 6 people each. All the tables and chairs have 4 legs each. Altogether there are 364 furniture legs. If there is one place set for each person who will attend the family reunion, how many guests is Laurel expecting?

12 The area of each square is 25 square units. What is the perimeter of this figure?

For further thought: How can you rearrange the squares (so that sides of the squares line up) to create another figure …

  • → with the same perimeter?
  • → with a perimeter less than 120 units?
  • → with a perimeter greater than 120 units?

13 Use the digits 0–9 only once each, so that each expression has the same value.

For further thought: Which single-digit whole numbers cannot be the value of all three expressions?

14 Denny has a collection of coins.

  • One-fourth of his coins are pennies.
  • One-sixth of them are quarters.
  • The number of nickels is 1.5 times the number of quarters.
  • The rest of the coins are dimes.

The total value is $4.32

How many of each coin does Denny have?

15 A nominating convention is occurring in a state that has 7 western precincts and 8 eastern precincts. Each precinct has 100 delegates. Three-fifths of the delegates in each western precinct support candidate A, with the rest supporting candidate B. Fifty-five percent of the delegates in each eastern precinct support candidate B, with the rest supporting candidate A. Once the votes of all delegates in all precincts are counted, which candidate will win the nomination?

16 Fontine has a photograph with dimensions 4 inches by 6 inches. She wishes to enlarge it to fit into a frame that has dimensions 18 inches by 24 inches. What is the largest factor by which she can enlarge the photograph so that it will fit the frame?

For further thought: Suppose she has three frames: 18" × 24", 16" × 20", and 22" × 30", and she enlarges the photograph by three different factors to best fit the three frames. In which frame does the enlarged photograph fill the highest percentage of the available area?

17 A typical gallon of paint covers 200 square feet of surface area. You would like to paint the walls of your bedroom, which is a 14 × 18-foot rectangle. There are two doors, which will not be painted, that each measure 7 × 2.5 feet. There is one window, which will not be painted, that is 4 × 6 feet. All the walls are 8 feet high. If paint is sold in one-gallon cans, each costing $22, how much do you have to spend to ensure that you can paint the entire bedroom?

For further thought: Wallpaper comes in rolls that are 2 feet wide and 16 feet long. What price does a roll of wallpaper need to be so that using wallpaper is less expensive than paint for the bedroom?

18 Find the proper fraction with a denominator less than or equal to 10 that has a value closest to each of the following: 0.23, 0.27, 0.88, 0.89.

For further thought: Which (if any) of your answers change if you allow denominators up to 12?

19 You use a lawnmower whose cutting area is 20 inches by 20 inches to mow a yard measuring 40 feet by 100 feet. If you mow in a north/south pattern, starting in the southwest corner as shown in the diagram below, how far will you have walked by the time you finish in the southeast corner? Round to the nearest foot.

20 In American football, a team can score in the following ways: 2 points for a safety, 3 points for a field goal, 6 points for a touchdown with no conversion, 7 points for a touchdown with a 1-point conversion, and 8 points for a touchdown with a 2-point conversion. How many different ways can a team score a total of exactly 24 points?

21 A local restaurant is offering a special on pizza. You must choose: one of three different crust styles, one of three different cheeses (or no cheese at all), and from zero to three of twelve possible toppings. How many different pizzas could you construct?

22 The ladder on a fire truck is 50 feet long and is mounted 5 feet above the ground. The maximum angle of elevation of the ladder is 60 degrees. Is this ladder able to reach a window that is at a height of 52 feet?

For further thought: What angle of elevation is required for the ladder to reach a window at the height of 54 feet?

23 A pump can inflate a spherical balloon to size of 8 inches in diameter in 20 seconds. Assuming that the flow of gas from the pump is constant, how much longer will it take to inflate the balloon to 10 inches in diameter? Round to the nearest second.

24 A certain board game uses two standard six-sided dice. Each player rolls the dice and advances a number of squares equal to the sum of the values on the dice. If the player rolls doubles (two dice with the same value) on the first roll, the player is allowed to roll again. Similarly, if he rolls doubles on the second roll, he is allowed a third roll. If he rolls doubles on the third roll, he receives a penalty. What is the probability that a player will receive such a penalty on any given turn?

25 The radioactive element Plutonium-239 decays exponentially and has a half-life of 24,360 years. In other words, it takes 24,360 years for half of any amount of Plutonium-239 to decay. How long will it take for 100 grams of Plutonium-239 to decay to 75 grams, rounded to the nearest year?

26 A standard deck of playing cards consists of 52 cards: 13 cards in each of four suits. If you draw 5 cards at random from the deck, what is the probability that all 5 will be of the same suit?

27 XYZ company estimates its monthly profits, P, as a function of its sales, S (both measured in dollars): P = 1.5 S − 7000. The marketing department estimates sales as a function of the unemployment rate, U, (measured in percent):

S=25000U0.5U2 +10

If the unemployment rate always falls in the range 2 ≤ U ≤ 12, for which unemployment rates will XYZ company have a positive profit?

28 Leonard is hungry and plans to eat a snack consisting of some peanut butter and some yogurt, but he is concerned about both nutrition and cost. He looked up the following nutrition facts: A one-ounce serving of his peanut butter contains 4 grams of protein, 3 grams of carbohydrates, and 94 calories. It costs 14 cents. A one-ounce serving of his yogurt contains 0.8 grams of protein, 6.9 grams of carbohydrates, and 38 calories. It costs 10 cents. Leonard insists that his snack contains at least two ounces of yogurt and at least one ounce of peanut butter. He wants to consume at least 10 grams of protein and no more than 30 grams of carbohydrates. He does not want to spend more than 70 cents. What amount of peanut butter and yogurt should Leonard eat to minimize the total number of calories he consumes while meeting all of his nutrition and cost requirements?

Steve Ingrassia,, is a retired high school math and computer science teacher in Chesterland, Ohio. He is interested in applying multiple techniques to solving cross-disciplinary, real-world problems.

Molly Rawding,, is a mathematics specialist/coach at Fiske Elementary School in Lexington, Massachusetts. She enjoys collaborating with teachers to develop students' mathematical understandings through visuals and puzzles.

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Supplementary Materials


Problems to Ponder provides 28 varied, classroom-ready mathematics problems that collectively span PK12, arranged in order of grade level. Answers to the problems are available online.

Mathematics Teacher: Learning and Teaching PK-12


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