MATH 5TH GRADE UNIT 4 Measurement and Data Geometry - Operations and Algebraic Thinking Number and Operations in Base Ten

Students will identify volume as an attribute of a solid figure. Students will explain that a cube with 1 unit side length is “one cubic unit” of volume. Students will explain a process for finding the volume of a solid figure by filling it with unit cubes without gaps and overlaps. Students will measure the volume of a hollow three-dimensional figure by filling it with unit cubes without gaps and counting the number of unit cubes. Students will use unit cubes to create two different rectangular prisms with one given volume. MAFS.5.MD.3.3 MAFS.5.MD.3.4

Volume is how much space something takes up Volume is how much space something takes up. Volume is measured in cube shaped units. 1 cubic inch 1 cu. in. 1 in 3

Volume is how much space something takes up Volume is how much space something takes up. Volume is measured in cube shaped units. 1 cubic centimeter 1 cu. cm 1 cm 3

Volume is how much space something takes up Volume is how much space something takes up. Volume is measured in cube shaped units. 1 cubic foot 1 cu. ft. 1 ft. 3

This is a solid figure. A box is a typical example of a three-dimensional – solid figure. Volume is the measure of the space inside the solid figure. A box has a width. A box has a length. A box has a height. Length x Width x Height = Volume cubic

This is a solid figure. It has a length of 8 units. width of 4 units This is a solid figure. It has a length of 8 units. width of 4 units. height of 6 units. W = 4 L = 8 H =6

When we measure the volume of a solid we use the term cubic When we measure the volume of a solid we use the term cubic. We can measure volume with a cube that has a length of “one cubic unit” of volume. To find volume of a cube we multiply: L x L x L = V so 1 x 1 x 1 = 1 cubic unit

We can use cubic units to find the volume of solid figures We can use cubic units to find the volume of solid figures. This figure has a length of 3. It has a width of 3. height of 1. L x H x W = V cubic

This solid figure is three times as high as the previous figure This solid figure is three times as high as the previous figure. Will it have three times the volume? Length = 3 Width = 3 Height = 3 L x H x W = V cubic

You will use a cubic unit to find the volume of each of the following solid figures. Remember: each cubic unit is 1 x 1 x 1 = 1 cubic

Plug in the L, W, H, to find the Volume of this solid Plug in the L, W, H, to find the Volume of this solid. L x H x W = V ___ x ___ x ___ = ___ cubic

Plug in the L, W, H, to find the Volume of this solid Plug in the L, W, H, to find the Volume of this solid. L x H x W = V ___ x ___ x ___ = ___ cubic

Plug in the L, W, H, to find the Volume of this solid Plug in the L, W, H, to find the Volume of this solid. L x H x W = V ___ x ___ x ___ = ___ cubic

Find the volume.

Find the volume.

Find the volume.

Find the volume.

Challenge. Find the volume by counting the unit cubes.

You can take a solid apart to find the volume. 6 cubic + = 24 cubic units

When finding the volume of a solid figure you can fill the solid with unit cubes. There can be NO gaps or overlaps. How many green unit cubes can fit into the gray box?

How many green unit cubes fit?

Find the volume.

Find the volume.

Find the volume.

Use what you see to finish filling the box. What is the volume?

What is the volume?

Find the volume of each.

Find the volume of each.

Given the volume of 12 unit cubes, what could a solid look like Given the volume of 12 unit cubes, what could a solid look like? This is one example. Can you draw another example.

Given the volume of 8 unit cubes, what could a solid look like Given the volume of 8 unit cubes, what could a solid look like? Can you draw two examples.

Given the volume of 15 unit cubes, what could a solid look like Given the volume of 15 unit cubes, what could a solid look like? Can you draw two examples.

Given the volume of 21 unit cubes, what could a solid look like Given the volume of 21 unit cubes, what could a solid look like? Can you draw two examples.

Given the volume of 24 unit cubes, what could a solid look like Given the volume of 24 unit cubes, what could a solid look like? Can you draw two examples.

Given the volume of 27 unit cubes, what could a solid look like Given the volume of 27 unit cubes, what could a solid look like? Can you draw two examples.

Students will relate finding the product of three numbers (length, width, and height) to finding volume. Students will use the formula for area to develop an understanding of volume. Students will relate the associative property of multiplication to finding volume. Students will calculate volume of rectangular prisms and cubes, with whole number edge lengths, using the formula for volume in real world and mathematical problems. MAFS.5.MD3.5

V = lwh or V = Bh Volume equals length times width times height or Volume equals base times height

Find the area of the Base by multiplying its length by its width. B = l x w then multiply the area of the base by the height V = B x h

( 3 x 3 ) is represented by the first layer of the box ( 3 x 3 ) is represented by the first layer of the box. ( 3 x 3 ) x 3 is represented by the number of 3 x 3 layers ( 3 x 3 ) + ( 3 x 3 ) + ( 3 x 3 ) = 9 + 9 + 9 = 27 9 represents the size/area of one layer

Give this one a try. ( _____ x _____ ) = first layer ( _____ x _____ ) x _____ is the number of _____ x _____ layers ( _____ x _____ ) + ( _____ x _____ ) = _____ + _____ = _____

Now it’s your turn. ( _____ x _____ ) = first layer ( _____ x _____ ) x _____ is the number of 3 x 5 layers ( _____ x _____ ) + ( _____ x _____ ) = _____ + _____ = _____

We can group our dimensions in any order we like We can group our dimensions in any order we like. This is the ASSOCIATIVE property of Multiplication at work! To associate with someone is to hang out with them. These numbers can switch friends any time they like and still get the right answer! V = lxwxh or V = lxhxw or V = wxhxl or V = hxwxl Give it a try. Plug in the dimensions and mix them up. L = 3 W = 4 H = 5

Give this one a try. ( _____ x _____ ) = first layer ( _____ x _____ ) x _____ is the number of _____ x _____ layers ( _____ x _____ ) + ( _____ x _____ ) = _____ + _____ = _____

Give this one a try. ( _____ x _____ ) = first layer ( _____ x _____ ) x _____ is the number of _____ x _____ layers ( _____ x _____ ) + ( _____ x _____ ) = _____ + _____ = _____

Give this one a try. ( _____ x _____ ) = first layer ( _____ x _____ ) x _____ is the number of _____ x _____ layers ( _____ x _____ ) + ( _____ x _____ ) = _____ + _____ = _____

Give this one a try. ( _____ x _____ ) = first layer ( _____ x _____ ) x _____ is the number of _____ x _____ layers ( _____ x _____ ) + ( _____ x _____ ) = _____ + _____ = _____

Give this one a try. ( _____ x _____ ) = first layer ( _____ x _____ ) x _____ is the number of _____ x _____ layers ( _____ x _____ ) + ( _____ x _____ ) = _____ + _____ = _____

Give this one a try. ( _____ x _____ ) = first layer ( _____ x _____ ) x _____ is the number of _____ x _____ layers ( _____ x _____ ) + ( _____ x _____ ) = _____ + _____ = _____

Give this one a try. ( _____ x _____ ) = first layer ( _____ x _____ ) x _____ is the number of _____ x _____ layers ( _____ x _____ ) + ( _____ x _____ ) = _____ + _____ = _____

Students will label appropriate units of measure for volume. Students will decompose a composite solid into non-overlapping rectangular prisms to find the volume of the solid by finding the sum of the volumes of each of the decomposed prisms. Students will determine a missing dimension of a rectangular prism given two dimensions and the volume. Students will generate possible dimensions of a rectangular prism when given the volume. Students will solve real world problems involving volume. MAFS.5.MD.3.5

Sometimes you will be given a COMPOSITE prism that you will need to decompose ( break apart ) to find the total volume. In this example you will find the volume of each section and add them together. This box has two separate rectangular prisms. See the red lines I drew for you? All you need to do is find the volume of each and add them together. The first prism is 10 x 4 x 6 The second prism is 4 x 6 x 8 Do the math and add them up! 4 V + V = total volume 8 10 4 6

Find the volume of each and add to find the total volume. Draw an imaginary line to decompose the composite prism Find the volume of each Add them together to find the total volume of the figure. _____ x _____ x _____ = ______ _____ + _____ = _______ is the total volume

Find the volume… First find the green. Then find the blue Find the volume… First find the green. Then find the blue. Add them together to find the total volume.

Find the volume.

Find the volume.

What is the total volume?

Find the volume.

Find the length of the present. H = 12 units W = 4 units L = ? V = 144 units cubic

Find the height of the tissue box. W = 8 units L = 2 units V = 64 units cubic

Find the width of the juice box. H = 5 units W = ? L = 4 units V = 40 unit cubic

Find the height of the cereal box. W = 3 units L = 10 units V = 210 units cubic

Find the width of the fish tank. H = 5 units W = ? L = 3 units V = 30 unit cubic

Find the width of the swimming pool Find the width of the swimming pool. Volume = 900 units cubic H = 6 units L = 15 units W = ?

How many possible solutions can you find to create a volume of: 30 ft ? 3 ? ?

How many possible solutions can you find to create a volume of: ? 3 ? ?

How many possible solutions can you find to create a volume of: 20 cm ? 3 ? ?

How many possible solutions can you find to create a volume of: 40 mm ? 3 ? ?

How many possible solutions can you find to create a volume of: 125 cm ? 3 ? ?

You can use the area formula to help you find the volume You can use the area formula to help you find the volume. Area = l x w Given the area of 14 feet squared: what would the volume be if the height is 5 feet? Base of object is 14 feet squared Height of object is 5 feet 14 x 5 = _____ feet cubic _____ ft 3

You can use the area formula to help you find the volume You can use the area formula to help you find the volume. Area = l x w Given the area of 20 cm What would the volume be if the height is 8 cm? Base of object is 20 cm sq Height of object is 8 cm = _____ cm cubic _____ cm 2 3

Area = 9 sq H = 7 V = ______ Area = 12 sq H = 3 Area = 25 sq H = 6

Students will compare and describe the geometric attributes of two-dimensional figures. Students will categorize two-dimensional figures according to their individual and shared geometric attributes. Students will explain the reasoning for determined categories. Students will select two-dimensional figures belonging to a given subcategory. MAFS.5G.2.3

3 We are all triangles. We all have three sides and three angles. TRI means three 3

These are all triangles. There are different types of triangles. All sides are equal. All angles are equal. No sides are equal. No angles are equal. Two sides are equal. Two angles are equal.

There is also a RIGHT Triangle. It has one angle that is 90 degrees.

Triangles can also be sorted by degree of each angle Triangles can also be sorted by degree of each angle. An angle that is less than 90 degrees is called ACUTE. The angle is little and “cute.” An angle that is more than 90 degrees is called OBTUSE. The angle is large and “obese.” obtuse angle

4 We are all quadrilaterals. Quadrilaterals all have four sides and four angles. QUAD means four. 4

Be careful! All four sided figures are QUADRILATERALS. 1 4 2 3 What about this? It has four sides. It has four angles. YES! It is a QUADrilateral!

When you are classifying shapes you must count the NUMBER OF SIDES. ARE THE SIDES PARRALLEL? like railroad tracks ARE THE SIDES PERPENDICULAR? like a crossroad ARE THE SIDES CONGRUENT? exactly the same

Parallel lines are two lines that are always the same distance apart and never touch.

Perpendicular means "at right angles". A line meeting another at a right angle, or 90° is perpendicular to it.

They have to be the same size and shape. CONGRUENT Exactly the same. They have to be the same size and shape. If you lay them on top of each other they fit perfectly!

Explain how these shapes were sorted.

Look at each shape. Which category do they belong in? Why?

Look again. Which category do they belong in? Why? angles congruent sides congruent parallel lines or perpendicular lines

These are examples of symmetrical design These are examples of symmetrical design. When something is symmetrical each part is exactly the same. Can you draw the lines of symmetry? I did one for you.

Symmetrical or not? Q and

Students will draw a coordinate plane with two intersecting perpendicular lines. Students will identify the intersection as the 0rigin and the point where 0 lies on each of the lines. zer0 = 0rigin Students will label the horizontal axis as the x-axis, and the vertical axis as the y-axis. X comes before Y in the alphabet = (x,y) Students will identify an ordered pair such as (3,2) as an x-coordinate followed by a y-coordinate. Students will explain the relationship between an ordered pair and its location on the coordinate plane. MAFS.5.G.1.1

four quarters or four quadrants. This is a COORDINATE PLANE. It is divided into four quarters or four quadrants.

It has two intersecting PERPENDICULAR This is a COORDINATE PLANE. It has two intersecting PERPENDICULAR lines. It has four intersecting angles of 90 degrees. 90 degrees

The point where the lines intersect is labeled 0. This is a COORDINATE PLANE. The point where the lines intersect is labeled 0. 0 is the 0rigin.

This is a COORDINATE PLANE. The HORIZONTAL line is the X axis. X

Y This is a COORDINATE PLANE. The VERTICAL line is the Y axis.

Remember: x comes before y in the alphabet. This is a COORDINATE PLANE. When you are listing POINTS on the coordinate plane you always list the X first and then the Y. ( X , Y ) Remember: x comes before y in the alphabet. X

Y X This is a Each line has been given a number. 5 4 3 2 1 This is a COORDINATE PLANE. Each line has been given a number. The number is the line’s name on the coordinate plane. X 1 2 3 4 5

Y X This is a COORDINATE PLANE. You find the POINT 5 4 3 2 1 This is a COORDINATE PLANE. You find the POINT on the coordinate plane by listing the line’s number on the X axis and then the line’s number on the Y axis. ( X , Y ) This is called an ORDERED PAIR. X 1 2 3 4 5

first comes X and then comes Y! 5 4 3 2 1 Can you find the point for this ordered pair? ( 2 , 3 ) Remember: first comes X and then comes Y! ( X , Y ) (2,3) X 1 2 3 4 5 Move right on the X axis to line 2. Then move up on the Y axis to line 3. Make a dot and label the ordered pair.

Find these ordered pairs on the coordinate plane. Y 5 4 3 2 1 Find these ordered pairs on the coordinate plane. ( 1 , 2 ) ( 5 , 2 ) ( 3 , 5 ) ( 1 , 2 ) Now, connect the points. What shape did you make? X 1 2 3 4 5

Find these ordered pairs on the coordinate plane. Y 5 4 3 2 1 Find these ordered pairs on the coordinate plane. ( 1 , 1 ) ( 1 , 4 ) ( 4 , 4 ) ( 4 , 1 ) ( 1 , 1 ) Now, connect the points. What shape did you make? X 1 2 3 4 5

Find these ordered pairs on the coordinate plane. Y 5 4 3 2 1 Find these ordered pairs on the coordinate plane. ( 1 , 2 ) ( 1 , 3 ) ( 5 , 3 ) ( 5 , 2 ) Now, connect the points. What shape did you make? X 1 2 3 4 5

and vertical movements from another point. Students will determine when a mathematical problem has a set of ordered pairs. Students will use appropriate tools strategically to identify, locate, and plot, ordered pairs of whole numbers on a graph in the first quadrant of the coordinate plane. Students will locate an unknown point on a coordinate plane when given horizontal and vertical movements from another point. Students will describe the horizontal and vertical movements necessary to get from one point to another on a coordinate plane. Students will relate the coordinate values of any graphed point to the context of the problem. Students will name or graph the point that would complete a specified, two-dimensional geometric shape in the first quadrant. MAFS.5.G.1.2

This is called the FIRST QUADRANT. It only has positive numbers. This is a compass rose. It shows cardinal directions of North, South, East, and West. You can use these directions to help you move or navigate on a coordinate plane. This is called the FIRST QUADRANT. It only has positive numbers.

Y X Draw a coordinate plane. Identify and label the 0rigin with a 0. Now find the X axis. Draw an X at the end of the line. Now fine the Y axis. Draw a Y at the top of the line. X

Y X 5 4 3 2 1 Draw a coordinate plane. Identify and label the 0rigin with a 0. Now find the X axis. Draw an X at the end of the line. Now fine the Y axis. Draw a Y at the top of the line. Now number the X axis. Now number the Y axis. X 1 2 3 4 5

Y 5 4 3 2 1 Now we will locate some ordered coordinates on the coordinate plane. Remember that the X is always listed before the Y. ( X , Y ) abcdefghijklmnopqrstuvwXYz - X comes before Y X 1 2 3 4 5

Y X Now try this set of coordinates. ( 2, 4 ) Locate the X axis. ( 2, 4 ) Locate the X axis. Find the line designated as 2. Locate the Y axis. Find the line designated as 4. Find the POINT where line 2 and line 4 meet. That is ( 2, 4 ) on the coordinate plane. 5 4 3 2 1 X 1 2 3 4 5

Y X The X coordinate tells you how far to travel to the East of the 0rigin The Y coordinate tells you how far to travel North of the 0rigin. Try this one. ( 3 , 1 ) Be able to explain your movements. “I traveled East on the X axis 3 and then I traveled North on the Y axis 1 to find point ( 3,1 ).” 5 4 3 2 1 X 1 2 3 4 5

Label the 0rigin as 0. Label the X axis and Y axis. Label the numbers on X. Label the numbers on Y. Using the cardinal directions of North, South, East, and West, to explain how you located this ordered pair. ( 2, 3 )

Label the coordinate plane. Locate the ordered pair of ( 5 , 3 ). Explain how you found that location.

Label the coordinate plane. Locate the ordered pair of ( 4 , 1 ). Explain how you found that location.

Label the coordinate plane. Locate the ordered pair of ( 3 , 5 ). Explain how you found that location.

Look at the square that is taking shape. Where on the coordinate plane does the next point need to be to complete the shape? ( 4,1 )

( , ) Look at the rectangle that is taking shape. Where on the coordinate plane does the point need to be to complete the shape? ( , )

( , ) Look at the trapezoid that is taking shape. Where on the coordinate plane does the point need to be to complete the shape? ( , )

( , ) Look at the octagon that is taking shape. Where on the coordinate plane does the point need to be to complete the shape? ( , )

Students will generate two numerical patterns with the same starting number for two given rules. Students will explain the relationship between the two numerical patterns by comparing how each pattern grows or by comparing the relationship between each of the corresponding terms from each pattern. Students will form ordered pairs out of corresponding terms from each pattern. Students will graph the ordered pairs out of corresponding terms Students will identify relationships between two patterns and use these relationships to make predictions or generalizations. MAFS.5.OA.2.3

Students will create a T chart or a Function chart with two numerical patterns with the same starting number for two given rules. P1 P2 ORDERED PAIRS ( 0,0 ) 1 4 ( 1,4 ) 2 8 ( 2,8 ) 3 12 ( 3,12 ) For the yellow triangle IN side of the Function chart the RULE is ADD 1. For the yellow star OUT side of the Function chart the RULE is ADD 4.

After finding the RULE for each pattern you need to see the relationship between the patterns. Pattern 1 is ADD 1 while Pattern 2 is ADD 4. The relationship between P 1 and P 2 is X 4. 0 x 4 = 0 1 x 4 = 4 2 x 4 = 8 3 x 4 = 12 P1 P2 ORDERED PAIRS ( 0,0 ) 1 4 ( 1,4 ) 2 8 ( 2,8 ) 3 12 ( 3,12 )

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Now that both Patterns are entered into the Function chart you can PLOT your POINTS on the coordinate plane. P1 P2 ORDERED PAIRS ( 0,0 ) 1 4 ( 1,4 ) 2 8 ( 2,8 ) 3 12 ( 3,12 ) Now connect the points! 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Look at the pattern. Can you guess what points will be next? ( , ) ( , ) ( , ) What trend does the pattern show? P1 P2 ORDERED PAIRS ( 0,0 ) 1 4 ( 1,4 ) 2 8 ( 2,8 ) 3 12 ( 3,12 ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Fill in the Function chart Fill in the Function chart. P 1: 8, 6, 4, 2, 0 P 2: 24, 12, 6, 3, 0 Find the patterns. What’s the Rule? P1 _________ P2 _________ Label the ordered pairs. Plot the points. Connect the points. Do you see a relationship? P1 P2 Ordered Pairs

Fill in the Function chart Fill in the Function chart. P 1: 0, 2, 4, 6 P 2: 0, 3, 9, 27 Find the patterns. What’s the Rule? P1 _________ P2 _________ Label the ordered pairs. Plot the points. Connect the points. Do you see a relationship? P1 P2 Ordered Pairs

Fill in the Function chart Fill in the Function chart. P 1: 0, 2, 6, 8, 10 P 2: 0, 4, 12, 16, 20 Find the patterns. What’s the Rule? P1 _________ P2 _________ Label the ordered pairs. Plot the points. Connect the points. Do you see a relationship? P1 P2 Ordered Pairs

Fill in the Function chart Fill in the Function chart. P 1: 3, 6, 9, 12, 15 P 2: 20, 15, 10, 5, 0 Find the patterns. What’s the Rule? P1 _________ P2 _________ Label the ordered pairs. Plot the points. Connect the points. Do you see a relationship? P1 P2 Ordered Pairs

Make predictions on the relationships in each graph.

Make predictions on the relationships in each graph.

Make predictions on the relationships in each graph.

Make predictions on the relationships in each graph.

Use the following ordered pairs to create a graph on your graph paper Use the following ordered pairs to create a graph on your graph paper. ( 6,1 ) ( 4,4 ) ( 5,6 ) ( 4,7 ) ( 6,9 ) ( 6,13 ) ( 8,13 ) ( 8,9 ) ( 9,9 ) ( 9,13 ) ( 11,13 ) ( 11,9 ) ( 13,7 ) ( 12,6 ) ( 13,4 ) ( 11,1 ) ( 6,1 ) Connect the points in order.

Now draw circles for eyes and a triangle nose. What did you make?