Computing and Problem Solving with Rational Numbers
Computing and Problem Solving with Rational Numbers
Objectives
Students will compute and solve problems using rational numbers. They will:
- add and subtract rational numbers using a number line.
- use properties of operations to compute with rational numbers.
- solve real-world problems involving rational numbers.
Essential Questions
How are relationships represented mathematically?
How can data be organized and represented to provide insight into the relationship between quantities?
How can expressions, equations, and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?
How can mathematics support effective communication?
How can probability and data analysis be used to make predictions?
How can recognizing repetition or regularity assist in solving problems more efficiently?
How does the type of data influence the choice of display?
How is mathematics used to quantify, compare, represent, and model numbers?
What does it mean to estimate or analyze numerical quantities?
What makes a tool and/or strategy appropriate for a given task?
- How is mathematics used to quantify, compare, represent, and model numbers?
- How are relationships represented mathematically?
- How can expressions, equations, and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?
- What makes a tool and/or strategy appropriate for a given task?
- How can recognizing repetition or regularity assist in solving problems more efficiently?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
NCTM’s applet, Equivalent Fractions, can be used to create equivalent fractions, by dividing and shading circles and squares.
IXL’s Grade 7 Add and Subtract Rational Numbers will give students additional practice with addition and subtraction of rational numbers.
IXL’s Grade 8 Add and Subtract Rational Numbers: Word Problems will give students additional practice with solving word problems that involve rational numbers.
IXL’s Grade 7 Multiply and Divide Rational Numbers offers students additional practice with multiplication and division of rational numbers.
IXL’s Grade 8 Multiply and Divide Rational Numbers: Word Problems offers students additional practice with solving word problems, involving rational numbers.
Formative Assessment
-
View
Multiple-Choice Items
- Which number line correctly models
?
- Which number line correctly models
?
- Last week, Harold used
gallons of gas. This week, he used 
gallons of gas. How many more gallons of gas did he use this week than last week?
- Jason ate
of a cake. Adam ate 
of the same cake. How much of the cake is left?
- −1.36 × 7.9 = ?
- −9.26
- −1074.4
- 926
- −10.744
- Roberto cuts 8 pieces of firewood, each of which is
feet in length. How many feet of firewood does he cut in total?
- 17
- 18
- Stephen paints
of a house. Chandler paints 
of what is left. How much of the house did Stephen and Chandler paint?
Multiple-Choice Answer Key
- C
- A
- D
- B
- B
- C
- D
- D
- C
Short-Answer Items:
10. Model
using a number line. State your answer.11. Andy needs
gallons of paint to finish a job. He only has 
gallons of paint. How many more gallons of paint does he need? Model the problem with an expression and solve it.12 Sandra plants
of the corn seeds. Arlan plants 
of what is left. What fraction represents the corn seeds that are left? Model the problem with a numeric expression and solve it.Short-Answer Key and Scoring Rubrics:
- Model
using a number line. State your answer.
A correct model is shown below:
The correct computation is
.Points
Description
2
- The student provides an accurate model and computation.
1
- The student provides an accurate model or computation.
0
- The student does not provide an accurate model or computation.
- Andy needs
gallons of paint to finish a job. He only has 
gallons of paint. How many more gallons of paint does he need? Model the problem with an expression and solve it.
A correct numeric expression is
. The correct answer is 
.Points
Description
2
- The student provides an accurate numeric expression and solution.
1
- The student provides an accurate numeric expression or solution.
0
- The student does not provide an accurate numeric expression or solution.
- Sandra plants
of the corn seeds. Arlan plants 
of what is left. What fraction represents the corn seeds that are left? Model the problem with a numeric expression and solve it
An accurate numeric expression is
. The correct answer is 
.Points
Description
2
- The student provides an accurate numeric expression and solution.
1
- The student provides an accurate numeric expression or solution.
0
- The student does not provide an accurate numeric expression or solution.
Performance Assessment:
- On Monday, Jonathan plans to mow
of a yard (lawn). On Tuesday, he plans to mow another 
of the same yard. Based on his plan, how much of the yard will he have left to mow after Tuesday? Write an expression that represents the solution. Solve. - Suppose Jonathan wishes to only have
of the yard left to mow after Tuesday. Write an expression that can be used to determine how much of the yard he would have to mow on Tuesday, if he still planned to mow 
of the yard on Monday. Solve. - Write an expression that verifies the accuracy of the fraction of the yard he will need to mow on Tuesday, based on the plan described in #2 above. Explain why the expression shows the solution to be correct.
- Jonathan remembers that he has an event on Tuesday, which will only give him time to mow another
of the yard. Considering that he still mows of the yard on Monday, determine how the fraction of the yard left to mow after Tuesday, based on this plan, compares to the fraction of the yard left to mow after Tuesday, from the plan, described in #2 above. Write an expression used to represent the difference and solve. - Jonathan’s father contemplates another plan. He thinks his son should mow of the yard on Monday and mow an equal fraction of the yard for each of the four remaining days in the week. Considering this plan, what fraction of the yard would Jonathan mow on Wednesday? Write an expression that can be used to solve the problem and show the solution.
- Due to Jonathan’s schedule, he ends up mowing
of the yard during the week. He mows an equal fraction of the yard during each of the days of Monday, Wednesday, and Friday. Write an expression that represents the fraction of the yard he mowed each of the three days. Compute.
Performance Assessment Answer Key:
1.
2.
3.
remaining 4.
He will have
more of the yard to mow.5.
Check:
6.
Check:
Performance Assessment Scoring Rubric:
Points
Description
4
- All answers are complete and correct.
- Work is shown and goes beyond the requirements of the problem.
- The response demonstrates advanced understanding of modeling word problems with expressions, solutions, and explanations for verifying the accuracy of solutions.
3
- Most answers are complete and correct.
- Work is shown for most problems, and all requirements of the problems are met.
- The response demonstrates a strong understanding of modeling word problems with expressions, solutions, and explanations for verifying the accuracy of solutions.
2
- Many answers are complete and correct.
- Little work is shown, and it fails to meet all the requirements of the problems.
- The response demonstrates a limited understanding of modeling word problems with expressions, solutions, and explanations for verifying the accuracy of solutions.
1
- Some answers are complete and correct.
- Very little correct work is shown.
- The response demonstrates a very limited understanding of modeling word problems with expressions, solutions, and explanations for verifying the accuracy of solutions.
0
- Very few or no answers are correct.
- No work is attempted, or the requirements of problems are not even partially met.
- The response demonstrates no understanding of modeling word problems with expressions, solutions, and explanations for verifying the accuracy of solutions.
- Which number line correctly models
Final 04/12/13
