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Math Trail (step 4) 1-5


Stop- 1

The horses pin is bordered by a wooden fence. The fence measures 60 ½ feet in length and 80 ½ feet in width. Using these dimensions, what would the perimeter of the fence be?

Bonus points:(1)- How much space does the horse have to run around? Find out what the area inside of the fence is.


Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.



Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.


Stop: 2

This is an old tractor used for many purposes on the Dolorosa Horse farm. If the gas tank on the tractor can hold 75 gallons, write a multiplication problem using fractions to determine how full the tank would be if it were 3/4th of the way full.

Bonus points:(2)- If gas costs $1.50 a gallon, how much would it cost to fill up the rest of the tank?


Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.


Stop: 3

One bale of hay costs the farmer $19. The farmer has 4 Stallions and 6 Mares.

In order to feed his horses, each horse must have one bail of hay per day. How much more money will the farmer spend on bails of hay to feed the Clydesdale horses?

Bonus:(2)- How much money would it cost the farmer to feed all of his horses for one week?


Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.



Stop: 4

One water bucket can hold 3 gallons of water. There are 62 water buckets on the Dolorosa Horse Farm. If all the buckets are full, how many gallons of water did the farmer use?

Bonus:(3)- The farmer wants to add 7 Foals to his horse farm. He will need to put 7 water buckets in their pin, but because Foals are a smaller breed, their water buckets only need 2 gallons of water in each. How many gallons of water will the farmer need to keep all of his horses hydrated?


Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.


Stop- 5

If one horse harness cost $15.00, how much would the farmer need to spend in order to buy 4 horse harnesses?

Bonus:(1)- If the farmer finds 34 more dollars in his piggy bank, how many horse harnesses could she add to her collection? Would there be any money left over? If yes, how much money would she have left?


Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.


  1. Arabian horse– The Arabian horse is a breed of horse that originated on the Arabian Peninsula. With a distinctive head shape and high tail carriage, the Arabian is one of the most easily recognizable horse breeds in the world.
  2. Bale of hay– Hay compressed into a block by a hay baler
  3. Foal- A horse that is one year old or younger
  4. Harness- A set of straps used to direct a horse
  5. Mare- A female horse
  6. Stallion– A stallion is a male horse that has not been gelded

Math Activity: Close to 100

Game Directions

  • “Close to 100” is played with a deck of 44 cards – four cards each of the digits 0-9, plus four wild cards. Each pair of players needs one deck; each individual player needs a score sheet.The point of the game is to create double-digit numbers that sum as close to 100 as possible. Each game has five rounds.For round 1, deal six cards to each player. Players choose any four of the cards to make two double-digit numbers that, when added, come as close as possible to a total of 100. Wild cards can be assigned any value. Players record their numbers and the total on a score sheet. The player’s score for the round is the difference between that total and 100. The used cards are discarded, and the two cards remaining in each hand are kept for the next round.
  • For rounds 2 to 5, deal out four cards to each player and repeat the steps in round 1.
  • At the end of five rounds, players total their scores. The player with the lowest total wins.
  • Scoring Variation
  • The rules of play remain the same; only the scoring is modified. If the payer’s total is above 100, the score is recorded as positive. If the score is below 100, the score is recorded as negative. The player with the grand total closest to zero after five rounds is the winner.


  1. 73+28= 101 [1]
  2. 90+10=100 [0]
  3. 59+40=99 [-1]
  4. 70+30=100 [0]
  5. 58+42=100 [0]
  6. Score = 0


  1. 75+26= 101 [1]
  2. 81+21=102 [2]
  3. 37+64=101 [1]
  4. 48+53=102 [2]
  5. 29+65= 94 [-6]
  6. Score = 0

Strategies used:

  • Conor- “The first game I didn’t have a strategy because I didn’t understand the game yet. There was a misunderstanding with the directions, I comprehended that we were supposed to make two 2 digit numbers who’s sum comes as close to 100 with out it going over. Therefor when I one of my numbers went over 100, we found ourselves buzzed, that was until we referred back to the scoring variation and resorted to that. The second game went much smoother. During this game my strategy was to continuously go above 100, and then in the last round go below 100, so that my scores  cancel each other out bring my total to zero.”
  • Katie- “As with Conor, I did not have a clear understanding of the directions in the first game. Though during the second game my strategy was to come up with sums who’s total was as close to 100 as possible. I used my numbers wisely issuing that I did not have two large or two small numbers left to be carried on to the next round. For my Wild card, I always chose a number that wasn’t close to one of the numbers in my hand, in order to save it for when if could be most beneficial.
  • We honestly enjoyed playing this game so much that we continued a few more games turing it into a contest.

Creative thinking in subtraction

Case 2 – Ann

Grade 6, October

When do I borrow? Is this where I cross out something? What if there is a zero in the column I want to take from? When do I stop borrowing?

1. In Ann’s case 2, examine the methods presented by Jason, Bert, Holly, and Joe. Apply their methods to this problem: 83-56.

Jason’s thinking: 

  • 83 – 56= ?
  • First subtract 50 from 83.
  • 83 – 56 = 33
  • Then subtract 3 (which is part of the 56)
  • 33 – 3= 30
  • Finally subtract the remaining 3 from the 30
  • 30- 3= 27
  1. Jason started on the left with the tens column.
  2. Then tried to make the number “friendlier”, by making the ones column a zero.
  3. For Jason, going from 83- 3= 80 and 33-3= 30 was easier than 33- 6= 27

Bert’s thinking:

  • 83 – 56= ?
  • 83 – 10 = 73
  • 73 – 10 = 63
  • 63 – 10 = 53
  • 53 – 10= 43
  • 43 – 10= 33
  • 33 – 3 = 30
  • 30 – 3= 27
  1. Bert subtracts 5 groups of 10, then subtracts 30 to bring his ones place to zero, then subtracts 3 more to get 27

Holly’s thinking:

  • 83 – 56 = ?
  • Take away 3 from both to make 80- 53
  • 80 – 10= 70
  • 70 – 10 = 60
  • 60 – 10= 50
  • 50 – 10= 40
  • 40 – 10= 30
  • 30- 3= 27
  1. Holly’s thinking is similar to Bert’s although she starts off with his last step.
3. What is it that children need to be able to understand about number in order to solve math problems in these ways?
  • If there wasn’t regrouping [as in 83-56] then “when the number is bigger” you don’t switch, and you add the answer to the final number instead of subtracting it.
4. In case 4, Emily writes that one of her second-graders, Ivan, invented a solution to a subtraction problem that was both new to her and confusing to figure out.

What Emily observed:

  • 52 – 28 = ?
  • Ivan’s first answer was 36
  • His friend got 24 and explains why Ivan is wrong
  • His friend: “20 away from 50 is 30, but you can’t take the 2 away from the 8; you have to take the 8 away from the 2 because the 8 is on the bottom.”
  • Ivan: “but you can’t take the 8 aways from the 2 because the 8 is too big.”

Ivan’s Thinking:

  • You take the 20 away from the 50 and get 30.
  • Then you take the 8 aways from the 2 which is -6.
  • Then you take the -6 away from the 30 and you get 24.
5. Does Ivan’s method resemble any of those presented in Ann’s case 2?


6. Emily declares this to be a time when she was learning mathematics by interacting with her students. What is your reaction to that?

Everyone you will ever meet knows something you don’t – Bill Nye

“Teaching for Mastery of Multiplication”

Overall this article discusses the importance of multiplication skills. It is vitally important for students to master these skills early on in order create a strong foundation to build upon. The article goes on to explain that students who are fluent with multiplication facts feel more confident and are more likely to participate in the activities. These students are also able to move smoothly through their assignments and word problems. Students who are not fluent with there multiplication facts stuggel with making progress due to not only the amount of time spent figuring out a problem, but also because of discouragement. So why are students still lacking multiplication-fact skills in upper elementary and even secondary grades? How should multiplication be introduced to students and taught effectively? Well there is an infinite number of variables which may affect a students learning. A few include though are not limited to, moving schools, family issues, a learning disability, classroom setting, and even the educators instructions.

A few strategies the article highlights are listed below:

  • Providing concrete experiences and semiconcrete representations prior to purely symbolic notations
  • Teaching rules explicitly
  • Introducing the concepts through problem situations and linking new concepts to prior knowledge
  • Providing mixed practice” (Gurganus, 2005).

The method which stood out most for me is the Rate strategy. This focuses on a value or distance that associated with a unit. The product is the total value or distance associated with all the units. Because this problem involves a number of measured units, it can be represented with a number line.

Wallace, A. H., & Gurganus, S. P. (2005).  Teaching for mastery of multiplication. Teaching Children Mathematics,  12(1) 26 – 33.

The Kissing Hand: Story problems Applied

The Kissing Hand by Ruth E Harper

This children’s book is about a young raccoon who is afraid to leave his mother while he goes to school. His mother assures him he will be ok and asks him to hold out his hand. She gives him a kiss on his palm and tells him to close it tight. “whenever you feel lonely and need a little loving from home, just press your hand to your cheek and think, ‘Mommy loves you. Mommy loves you.'”

Below are four word problems based off of the story The Kissing Hand. Through addition and subtraction, students calculate the number of kisses on Chester’s hand, the number of berries picked on the way to school and the hours he has left at school.


  • Chester’s mom gave him twenty-five kisses this morning. He used fifteen kisses this afternoon. How many kisses are left?
  • Chester has been at school for 3 hours. If school lats for 7 hours, how many hours of school does Chester have left?


  • Chester has 3 kisses on his right hand and 5 kisses on his left hand. How many kisses are on Chester’s hands altogether?

    Chester picked 12 berries on his way to school this morning. Squirrel picked 8 more berries than Chester. How many berries does Squirrel have?

Case Studies

Class Readings – Blog Posting

The attached case studies present students’ thinking about the spoken and written number systems and the connections between them.

As you are reading, refer to the questions listed in the introductory text on page 44. Keep those questions and your reactions to them in mind as you respond to the following:

  1. Turn to Dawn’s case 11 and study Andrew. Why does it make sense to him to have “5 and 10” follow 59? What does he understand? What is he missing?
  2. In Danielle’s case 15, the children came up with many ways to write “one hundred ninety-five.” What sense do you see in each one?
  3. In Muriel’s case 14, the children talk about different kinds of zeros. Explain what they mean by this.
  4. Turn to Donna’s case 12. Use cubes or counters to do the bean-counting activity Donna describes. What mathematics is highlighted as you do this work?

You will need to record your responses to these questions on your blog prior to class on Wednesday.