1.7: Problem Solving Strategies: Guess, Check and Revise; Use Mental Math
Introduction
The Elephants Weigh In
There are two elephants at the city zoo, and they are also two different kinds of elephants. One is an African Elephant and the other is an Indian Elephant.
An African elephant is larger than an Indian elephant.
One of the fun jobs that city zookeepers get to do is to weigh in the elephants. It is always interesting to see how much each elephant weighs.
Tara Jonsen gets the fun job of weighing Jojo, a male African Elephant and Junas, an Indian Elephant. She wonders if just this once Junas will weigh more than Jojo.
Jojo weighs 4,000 pounds more than Junas.
Their combined weight is 26,000 pounds.
Tara leads them both back to their habitats. When she returns to the log book, she realizes that she forgot to write down each specific weight. She remembers two things.
That Jojo weighs 4,000 pounds more than Junas.
That their combined weight was 26,000 pounds
Given this information, can Tara figure out what each elephant weighed?
In this lesson, you will learn how to help Tara to figure this out using a couple of different strategies.
By the end of the lesson, you will know what each elephant weighed.
What You Will Learn
In this lesson, you will learn the following problem solving strategies:
- How to read and understand a given problem situation
- How to develop and use the strategy: Guess, Check and Revise
- How to develop and use the strategy: Use Mental Math
- Plan and compare alternative approaches to solving problems
- Solve real-world problems using selected strategies as part of a plan.
Teaching Time
I. Read and Understand a Given Problem Situation
In our last lesson, we worked on reading and understanding a given problem situation. We used this first step of the four-part problem solving plan as we worked with Tyler and his orangutan adoption problem.
Now we are going to apply this first step to the elephant problem.
Let’s look at the problem once again so that we can determine the given information and identify what information we are looking for.
Here is the problem.
There are two elephants at the city zoo, and they are also two different kinds of elephants. There is an African Elephant and an Indian Elephant.
An African elephant is larger than an Indian elephant.
One of the fun jobs that city zookeepers get to do is to weigh in the elephants. It is always interesting to see how much each elephant weighs.
Tara Jonsen gets the fun job of weighing Jojo, a male African Elephant and Junas, an Indian Elephant. She wonders if just this once Junas will weigh more than Jojo.
Jojo weighs 4,000 pounds more than Junas.
Their combined weight is 26,000 pounds.
Tara leads them both back to their habitats. When she returns to the log book, she realizes that she forgot to write down each specific weight. She remembers two things.
That Jojo weighs 4,000 pounds more than Junas.
That their combined weight was 26,000 pounds
Given this information, can Tara figure out what each elephant weighed?
Let’s underline all of the important information.
Our given information is:
Jojo weighs 4,000 pounds more than Junas.
Their combined weight is 26,000 pounds.
To understand this problem, we need to figure out two unknowns.
We need to figure out what Junas weighed and what Jojo weighed.
There is a relationship between the two weights.
II. Guess, Check and Revise
We can work on figuring out the weights of the two elephants by using guess, check and revise.
Guess, check and revise has us guess numbers that we think might work and try them out.
Since we don’t know a lot about what the two elephants weighed, this is probably a good strategy for this problem.
Jojo-let’s call his weight \begin{align*}x\end{align*}
Junas-let’s call his weight \begin{align*}y\end{align*}
\begin{align*}x + y = 26,000\end{align*}
Here is an equation that represents our problem.
Let’s guess a few numbers that might work in this problem.
What if Junas weighed 10,000 pounds?
We can say that Junas weight, \begin{align*}y+ 4000 =x\end{align*}, Jojo’s weight.
Here is our new equation. Let’s see if it works.
\begin{align*}10,000 + 4000 = 14,000 & = \text{Jojo's weight}\\ 10,000 & = \text{Junas weight}\\ 14,000 + 10,000 & = 24,000\end{align*}
Uh oh, our number is too small.
We need to revise. We could keep guessing numbers until we find ones that work.
Maybe it makes more sense to use some mental math.
III. Use Mental Math
We can use mental math to solve this problem.
If we take the total amount of weight, 26,000 pounds, and subtract 4,000 since that is the difference between the two elephants, we get a new answer.
22,000 pounds
We can next divide it in half for the two elephants.
\begin{align*}22,000 \div 2 = 11,000\end{align*}
That is the weight if the elephants were equal.
But one weighs more than the other so we can add 4,000 to 11,000.
Jojo weighs 15,000 pounds
Junas weighs 11,000 pounds
IV. Plan and Compare Alternative Approaches to Solving the Problem
Wow, we just used two completely different methods.
We can find the answer with whichever one we choose.
By comparing them we can conclude the following:
- Guess, check and revise can get you started but you may need to try several different options to get the correct answer.
- Mental math requires you to think in terms of the divisibility of numbers or multiples.
Now we can help Tara find the solution to her dilemma.
Real Life Example Completed
The Elephants Weigh In
Using mental math seemed to be a quicker solution to our elephant problem.
Let’s look at the problem once again.
There are two elephants at the city zoo, and they are also two different kinds of elephants. There is an African Elephant and an Indian Elephant.
An African elephant is larger than an Indian elephant.
One of the fun jobs city zookeepers get to do is to weigh in the elephants. It is always an interesting time to see how much each elephant weighs.
Tara Jonsen gets the fun job of weighing Jojo, a male African Elephant and Junas, an Indian Elephant. She wonders if just this once Junas will weigh more than Jojo.
Jojo weighs 4,000 pounds more than Junas.
Their combined weight is 26,000 pounds.
Tara leads them both back to their habitats. When she returns to the log book, she realizes that she forgot to write down each specific weight. She remembers two things.
That Jojo weighs 4,000 pounds more than Junas.
That their combined weight was 26,000 pounds
Given this information, can Tara figure out what each elephant weighed?
Here is our arithmetic.
\begin{align*}26,000 - 4,000\end{align*} difference between the weights \begin{align*}= 22,000\end{align*}
\begin{align*}22,000 \div 2\end{align*} for the two elephants \begin{align*}= 11,000\end{align*} pounds each
Jojo weighs 4,000 pounds more \begin{align*}= 11,000 + 4,000 = 15,000\end{align*} pounds
Junas weighs 11,000 pounds
Their total weight is 26,000 pounds.
Time to Practice
Directions: Use one of the problem solving plans that we covered in the last two lessons to solve the following problems. You may choose from the four-part problem solving plan, guess check and revise, or mental math. Be sure to write which plan you used and the answer.
1. Dana caught twenty-eight fish. She wants to divide the fish into four baskets. If she does this, how many fish will be in each basket? Can she put the same number of fish in each basket?
2. Carl also went fishing. He caught five fish on the first day and four fish on the next day. If he continues this pattern on what day will he not catch any fish?
3. Jessie loves to cook fish after she catches them. She is having ten people over for dinner. If each person eats a half of a fish, how many fish will she need to cook to feed all ten people?
4. Cass takes people out on a fishing boat to go deep sea fishing. With his strategies, people often catch double the amount of fish that they do regularly. If someone normally catches three fish in a day, how many fish will they catch using Cass’ strategy?
5. If you were someone who usually caught thirty-five fish in one week, how many fish would you catch on average per day?
6. Travis lives in Florida and loves going to pick strawberries during strawberry season. He can fit 25 strawberries in one box. If he is given a barrel of 500 strawberries, how many boxes will it take for Travis to sort the strawberries?
7. If he fills 45 boxes during his shift at work, how many strawberries did he start with?
8. If Travis works two shifts, at this rate, how many boxes will he fill?
9. How many strawberries did he sort over both shifts?
10. Josie helps tag the strawberries. She tags them at \begin{align*}\$2.00\end{align*} per pint. If she sells 100 pints, how much money has she made?
11. If Josie sells $400.00 worth of strawberries, how many pints has she sold?
12. Josie also enjoys making strawberry milkshakes. If it takes 5 strawberries to make one milkshake, how many can she make with 20 strawberries?
13. If Josie makes 35 strawberry milkshakes in one day, how many strawberries does she need to accomplish this task?
14. If there are 25 strawberries in a pint, how many pints does Josie use to make her 35 milkshakes?
15. Carl loves to collect old vinyl records. He has a whole collection that he received from his Dad. If he has five different categories of records with twenty records in each category, how many records does Carl have altogether?
16. Julie is a friend of Carl’s. She brought over her collection of records. Julie has 254 records. If she and Carl were to combine their collections, how many would they have altogether?
17. When Carl and his Mom went to a yard sale, Carl got a box of vinyl records for $25.00. He brought them home and looked in the box. Out of 30 records, five of them were broken or scratched. If he puts these new records with his collection, how many does he now have?
18. Carl’s sister borrowed ten records to show her friend. How many are left in Carl’s collection?
19. Mario is an outstanding skateboarder. He recently purchased a new skateboard. He wants to sell his old one. A friend wishes to buy it for \begin{align*}\$45\end{align*}. If he gives Mario three twenty dollar bills, how much change should Mario give his friend?
20. If Mario buys a new skateboard for double the price that he sold his old one, how much did he pay for the new skateboard?
21. If Maria has \begin{align*}\$100.00\end{align*} and he buys the skateboard for double the price that he sold his old one, does he have enough money to make the purchase?
22. Did he receive any change back at the skateboard shop? How much?
23. Karen collects fairy figurines. She was given 3 for her birthday, 2 for Christmas, 4 from her grandmother and 3 she bought on her own. How many fairy figurines does she have in all?
24. Karen’s little sister loves one of the figurines. Karen has decided to give her the little fairy as a gift. After she does this, how many figurines will Karen have left?
25. At a yard sale, Karen bought 5 fairies for \begin{align*}\$20.00\end{align*}. How much did she pay per fairy?
26. How many fairies does Karen have now?
27. Karen’s friend Emily also collects fairies. If Emily has twice as many fairies as Karen, how many does she have?
28. Jamie runs track at school. He is one of the fastest runners on the team and runs one mile in about 5 minutes. How long will it take Jamie to run 10 miles?
29. If Jamie runs a 3 mile race, about how how much time will it take to run the 3 miles at his one mile pace?
30. If Jeff runs his mile one minute slower than Jamie does, how long will it take Jeff to run the ten miles?