Post on 11-Mar-2018
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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G7-Module 3: Topic A Lesson 1: Generating Equivalent Expressions
Example 1
a.) Rewrite and by combining like terms.
Write the original expressions and expand each term using addition. What are the new expressions equivalents?
b.) Find the sum of and .
Example 4
f. Alexander says that is equivalent to because of any order, any grouping. Is he correct? Why
or why not?
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 3: Writing Products as Sums and Sums as Products
Example 1 Space for Drawings
a.) Draw a tape diagram representing the sum
using squares for units.
b.) Redraw the diagram without squares, as two rectangles
each with a height of 1 unit.
c.) Draw a rectangular array for .
Write the product as a sum: ________________________________
d.) Draw a tape diagram representing the sum
using square units.
e.) Redraw the diagram without squares, as two rectangles
each with a height of 1 unit.
f.) Draw a rectangular array representing
Write the product as a sum: __________________________________
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Example 6:
A number of square tiles are needed to border a square fountain with side length feet. Express the total
number of tiles needed in terms of three different ways.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 4: Writing Products as Sums and Sums as Products
Example 2
and are positive integers. , , and stand for the number of unit squares in a rectangular array.
Lesson 6: Collecting Rational Number Like Terms
Example 4:
Model how to write the expression in standard form using rules of rational numbers.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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G7-Module 3: Topic B Lesson 7: Understanding Equations
Example 1
The ages of three sisters are consecutive integers. The sum of their ages is . Find their ages.
a.) Use a tape diagram to find the situation.
b.) If represents the age of the youngest sister, write an equation that can be used to find the age of the
youngest sister and the other sister’s ages.
c.) Determine if your answer from part (a) is a solution to the equation you wrote in part (b).
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 8: Using the If-Then Moves in Solving Equations
Example 1
Julia, Keller, and Isreal are volunteer firefighters. On Saturday the volunteer fire department held its annual coin drop
fundraiser at a street light. After one hour, Keller had collected more than Julia and Isreal had collected less
than Keller. Altogether the three firefighters collected . How much did each person collect?
Find the solution using an arithmetic approach. Identify which key If-Then Moves are used.
The amount of money Julia collected is . Write an expression in terms of to represent the amount of money Keller
collected in dollars.
Using the expressions for Julia and Keller, write an expression to represent the amount of money that Isreal collected in
dollars.
Using the expressions above, write an equation in terms of that can be used to find the amount each person collected.
Solve the equation written above to determine the number of money each person collected and describe the If-Then
moves.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Example 1 (continued)
In terms of Julia Collected Isreal Collected Keller Collected
Julia’s Amount
Isreal’s Amount
Keller’s Amount
Lesson 9: Using the If-Then Moves in Solving Equations
Example 2
Shelby is seven times as old as Bonnie. If in 5 years, the sum of Bonnie and Shelby’s ages is 98, find Bonnie’s present age.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 10: Angle Problems and Solving Equations
Example 4
Two lines intersect in the following figure. The ratio of the measurements of
the obtuse angle to the acute angle in any adjacent angle pair is 2:1.
In a complete sentence, describe the angle relationships in the diagram.
Label the diagram with expressions that describe this relationship. Write an equation to that models the angle
relationship and solve for . Find the measurements of the acute and obtuse angles.
Lesson 11: Angle Problems and Solving Equations
Exercise 2
In a complete sentence, describe the angle relationships in the diagram.
Write an equation for the angle relationship shown in the figure and solve
for x and y. Confirm your answers by measuring with a protractor.
x°2x°
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 12: Properties of Inequalities
Station 1
Die 1 Inequality Die 2 Operation New Inequality Inequality Symbol Preserved or Reversed?
Add
Preserved
Add
Subtract
Subtract
Add
Examine the results. Make a statement about what you notice and justify it with evidence.
Station 2
Die
1 Inequality
Die
2 Operation New Inequality Inequality Symbol Preserved or Reversed?
Multiply
by -1
Reversed
Multiply
by -1
Multiply
by -1
Multiply
by -1
Multiply
by -1
Examine the results. Make a statement about what you notice and justify it with evidence.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Station 3
Die 1 Inequality Die 2 Operation New Inequality Inequality Symbol Preserved or Reversed?
Multiply
by
( ) ( )
Preserved
Multiply
by
Divide by
Divide by
Multiply
by
Examine the results. Make a statement about what you notice and justify it with evidence.
Station 4
Die 1 Inequality Die 2 Operation New Inequality Graph
Multiply
by
Multiply
by
Divide by
Divide by
Multiply
by
Examine the results. Make a statement about what you notice and justify it with evidence.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 13: Inequalities
Opening Exercise
Tarik is trying to save $265.49 to buy a new tablet. Right now he has $40 and can save $38 a week from his allowance.
Write and evaluate an expression to represent the amount of money saved after:
2 weeks
3 weeks
4 weeks
5 weeks
6 weeks
7 weeks
8 weeks
When will Tarik have enough money to buy the tablet?
Write an inequality that will generalize the problem.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 14: Solving Inequalities
Example 1
A youth summer camp has budgeted for the campers to attend the carnival. The cost for each camper is
which includes general admission to the carnival, unlimited rides, and meals. The youth summer camp must also pay
for the chaperones to attend the carnival and for transportation to and from the carnival. What is the
greatest amount of campers that can attend the carnival if the camp must stay within their budgeted amount?
Lesson 15: Graphing Solutions to Inequalities
Example 1
A local car dealership is trying to sell all of the cars that are on the lot. Currently it has cars on the lot and the
general manager estimates that they will sell cars per week. How long will it take for the number of cars on the lot to
be less than ?
Write an inequality that can be used to find the number of weeks.
Solve and graph the inequality.
Interpret the solution in the context of the problem.
Verify the solution.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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G7-Module 3: Mid-Module Assessment
MMA Problem #6
6. Jenny invited Gianna to go watch a movie with her family. The movie theater charges one rate for 3D admission and a different rate for Regular Admission. Jenny and Gianna decided to watch the newest movie in 3D. Jenny’s mother, father, and grandfather accompany Jenny’s little brother to the Regular Admission movie. a. Write an expression for the total cost of the tickets. Define the variables.
b. The cost of the 3D ticket was double the cost of the regular admission. Write an equation to represent the relationship between the two types of tickets.
c. The family purchased refreshments and spent a total of . If the total amount of money spent on tickets and refreshments was use an equation to find the cost of one regular admission ticket.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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G7-Module 3: Topic C Lesson 16: The Most Famous Ratio of All
Opening Exercise:
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 17: The Area of a Circle
Discussion
What do the longer sides of the resulting rectangle correspond to on the circle?
What does the height of the rectangle correspond to on the circle? How do you know?
Formalize the relationships between the area of the rectangle and the area of a circle.
Example 1
Michael is laying tile in the floor of a square walk-in shower. He wants a circle pattern as shown. He needs to determine
the area of the circle so that he can order the special tile for the shower project. Michael drew the tile pattern on graph
paper so that he could have a visual model to help guide his thinking.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Problem Set #10
If the diameter of a circle is double the diameter of a second circle, what is the ratio of area of the first circle to the area
of the second?
Lesson 18: More Problems on Area and Circumference
Opening Exercise:
Draw a circle of diameter 12 cm and a square of side length 12 cm on grid paper. Determine the area of the square and
the circle.
Example 3:
The circumference of a circle is . What is the exact area of the circle? Draw a diagram to assist you in
solving the problem.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 19: Unknown Area Problems in the Coordinate Plane
Example 1: Area of a Parallelogram
The coordinate plane below contains figure , parallelogram .
1. Write the ordered pairs of each of the vertices next to the vertex
points.
2. Draw a rectangle surrounding figure that has vertex points of and
. Label the two triangles in the figure as and .
3. Find the area of the rectangle.
4. Find the area of each triangle.
5. Use these areas to find the area of parallelogram .
The coordinate plane below contains figure , a rectangle with the same base as the parallelogram above.
6. Draw triangles and next to so that you have a rectangle that is the
same size as the one you created on the first coordinate plane.
7. Find the area of rectangle R.
8. What do figures R and P have in common?
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Problem Set #11
Find the unknown value labeled as on the figure.
11. The trapezoid has an area of sq. units.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 20: Composite Area Problems
Example 3:
Find the area of the shaded region.
Problem Set #5
5. Find the area of region which is formed by three semicircles.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 21: Surface Area
Example 1
A right triangular prism , a right rectangular prism, and a right pentagonal prism are
pictured below and all have equal heights of .
a.) Write an expression that represents the lateral area of the right triangular prism
as the sum of the areas of its lateral faces.
b.) Write an expression that represents the lateral area of the right rectangular prism
as the sum of the areas of its lateral faces.
c.) Write an expression that represents the lateral area of the right pentagonal prism
as the sum of the areas of its lateral faces.
d.) What value appears often in each expression and why?
e.) Rewrite each expression in factored form using the distributive property and the
height of each lateral face.
f.) What do the parentheses in each case represent with respect to the right prisms?
h
g.) How can we generalize the lateral area of a right prism into a formula that applies
to all right prisms?
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 22: Surface Area
Exit Ticket
1. Find the surface area of the pyramid.
2. Six cubes are glued together to form the solid shown in the
diagram. If the edges of each cube measure inches in
length, what is the surface area of the solid?
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 23: The Volume of a Right Prism
The Volume of a Right Triangular Prism
How could we create a right triangular prism with five times the volume of the right triangular prism pictured
below, without changing its base? Draw your solution on the diagram, then give the volume of the solid.
Exit Ticket
The base of the right prism is a hexagon composed of a rectangle and two
triangles. Find the volume of the right hexagonal prism using the formula
.
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 24: The Volume of a Right Prism
Example 3
of water are poured into a container in the shape of a
right rectangular prism. The inside of the container is
long, wide, and tall. How far from the top of the
container is the surface of the water? ( )
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 25: Volume and Surface Area
Exercise 1: Fish Tank Designs
Two fish tanks are shown below.
a. Which tank is holds the most water?
b. The cost of a fish tank depends on the volume and the surface area. Which tank
would be the ‘best buy’? Explain your reasoning.
c. The pet store advises no more than inch of fish per gallon of water. Ray has a inch
angel fish, three inch tetras, and a inch catfish. How many more inches of
fish will fit into the larger of the two tanks?
d. Water evaporates at a rate of cubic inches per day. What will be the height of
the water in each tank after one week?
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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Lesson 26: Volume and Surface Area
Exercise 1: Designing a Brick Planter
You have been asked by your school to design a brick planter that will
be used by classes to plant wild flowers. The planter will be in the
shape of a right rectangular prism and the external dimensions are to
be ft. ft. ft. The planter base and sides will be made of a
layer of bricks that are inches thick.
a. What are the internal dimensions of the planter?
b. If you are going to fill the planter ¾ full of soil, how much soil will you need to
purchase and what will be the height of the soil?
c. What is the volume of bricks that will be used to construct the planter?
d. If brick has a volume of in in in, how many bricks are needed for the
planter (round to the nearest hundred)?
e. If soil is sold in cubic-foot bags that cost /bag and bricks are sold in stacks of
for /stack, how much will it cost to construct the planter?
NYS COMMON CORE MATHEMATICS CURRICULUM 7•3 Student Examples and Exercises
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End-of-Module Assessment Task – Question #3
3. A new park was designed to contain two circular gardens. Garden A has a diameter of m, and the garden B has a diameter of m. a. If the Gardner wants to outline the gardens in edging, how many meters will be needed to outline
the smaller garden? (Write in terms of .)
b. How much more fencing will be needed for the larger garden than the smaller one? (Write in terms of .)
c. The Gardner wishes to put down weed block fabric on the two gardens before the plants are planted in the ground. How much fabric will be needed to cover the area of both gardens? (Write in terms of .)