Download What`s Included - Lakeshore Learning

Designed to meet these objectives:
Math
• Students will construct and solve number sentences.
• Students will understand the inverse relationships of operations.
• Students will understand the associative, commutative, and identity properties for addition and
multiplication.
• Students will compare fractions and identify equivalent fractions.
• Students will perform operations with fractions using concrete models.
• Students will explore probability using spinners and predict outcomes.
• Students will describe and compare 2-dimensional shapes and 3-dimensional shapes.
• Students will understand that nets represent 3-dimensional shapes.
• Students will measure perimeter and area by using grids and by making appropriate calculations.
• Students will use protractors to measure angles.
• Students will identify equivalent percents, decimals, and fractions.
• Students will plot points on a coordinate grid.
• Students will evaluate algebraic expressions involving variables and graph functions on a coordinate grid.
Turn your overhead projector into an exciting math center with this all-inclusive kit! Sturdy, transparent plastic pieces make it easy to provide concrete representations of abstract math concepts to the entire class.
You’ll find activity suggestions and teaching tips for everything from operations to fractions, geometry, and
algebra in this handy guide.
What’s Included
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92 number and operation tiles
22 fraction arrays
30 shapes (2-D)
Protractor
Decimal boards
Coordinate plane
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17 fraction circles (101 pieces total)
6 solid shapes (3-D)
14 nets (3-D)
Spinner
+/– number lines
Perimeter and area graphing board
Note: Some materials are protected with a transparent film during shipping. Be sure to peel away the film
before using the materials.
©2009 Lakeshore
(800) 428-4414
www.lakeshorelearning.com
FF701
Ages 8+
Printed in China
Exploring Operations
Materials: Number and operation tiles
Open-Number Sentences
Display open-number sentences such as “22 + ___ = 58” or “60 ÷ ___ = 12.” Then, challenge students to
find numbers that make the equations true. When they are comfortable with solving problems in this way,
introduce more complex equations involving mixed operations such as “22 + ___ = 58 – 26” or inequalities
(comparisons) such as “60 ÷ ___ > 20.”
Inverse Operations
To help children discover the relationship between “x” and “÷,” place tiles for 45, 15, 3, and = on the overhead projector. Challenge students to add an operation tile to form an equation. In this example, students
could use the ÷ tile to create “45 ÷ 15 = 3” or the x tile to create “45 = 15 x 3.” Explain that multiplication
and division are inverse, or opposite, operations. Ask if anyone knows the inverse operation for addition.
Display “26 + 14 = 40” and help children determine that its inverse can be either “40 – 26 = 14” or
“40 – 14 = 26.” Help students conclude that the inverse of addition is subtraction.
Associative Property
Write an equation such as “(15 + 25) + 35 = 75” on the transparency. Remind students that the parentheses
tell us to solve that part of the equation first. Does the problem “15 + (25 + 35)” have the same solution?
Explain that this is an example of the associative property of addition: “When three or more numbers are
added, changing the way the numbers are grouped does not affect the sum.” Does the associative property
also apply to multiplication? Demonstrate with an equation such as “(3 x 5) x 4 = 60.” If the equation is
rewritten as “3 x (5 x 4) = 60,” is it still true? Finally, point out that the associative property does not apply
to subtraction or division.
Commutative Property
Display an equation such as “78 + 7 = 85.” How can these tiles be rearranged to form another equation that
equals 85? (7 + 78 = 85) Explain that this is an example of the commutative property of addition: “The
order in which numbers are added does not affect the sum.” Does the commutative property also apply to
multiplication? Demonstrate with an equation such as “33 x 2 = 66.” If the tiles are rearranged to form
“2 x 33 = 66,” is the equation still true? Finally, point out that the commutative property does not apply to
subtraction or division.
Identity Properties
Use the tiles to display a problem such as “37 + 0.” What is the sum? (37) Repeat with other examples, such
as “50 + 0” or “0 + 2.” Explain that these are examples of the additive identity property (also known as the
addition property of zero): “When zero is added to any number, the sum is the same as the number.” Ask,
“Does this work for multiplication, too?” (No, multiplying a number by 0 results in 0.) Explain that there is a
similar property called the multiplicative identity property, or the multiplication property of one: “When a
number is multiplied by one, the product is equal to the original number.”
Understanding Fractions
Materials: Fraction circles, fraction arrays, operation tiles (>, <, and =)
Comparing Fractions
Use the fraction circles to help students compare fractions such as 3⁄4
and 3⁄6. Write the two numbers on a blank overhead sheet, and then
invite volunteers to display the fractions by placing the fraction
pieces above the numbers on the sheet. Which fraction is bigger?
Add the appropriate operation tile to complete the equation (3⁄4 > 3⁄6).
Help students discover that, for fractions with equal numerators, the
fraction with the smaller denominator is always the larger number.
Comparisons can be practiced using the fraction arrays. Explain that
the shaded portion of each array represents the numerator of a
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fraction and the total number of sections represents the denominator. Display arrays representing fractions
such as 1⁄2, 1⁄3, 1⁄4, and 1⁄5, and have students sequence them from smallest to largest. (Encourage students to
use the fraction circles to check their work.) To increase the difficulty, have students compare pairs such as 2⁄3
and 4⁄5, using operation tiles (<, >, or =) to complete the equations.
Equivalent Fractions
Display the halves and fourths fraction circles on your overhead projector, on top of a blank overhead sheet.
Trace and then remove one of the half pieces and challenge students to use other pieces to fill the empty
space. Help students discover that two fourths take up the same space as one half, and model how this can
be written as an equation (2⁄4 = 1⁄2). Challenge students to find other fraction pieces that equal 1⁄2, and have
them write the equations. Then, help them find other fractional equivalences, such as “2⁄6 = 1⁄3” or “9⁄12 = 3⁄4.”
Adding and Subtracting Fractions
• Use a blank transparency as a background and write a
problem such as “1⁄2 + 1⁄4.” Have volunteers use the fraction
circles to show the problem. Explain that, in order to add
fractions, you must have a common denominator. Model
how to exchange 1⁄2 for its equivalent, two 1⁄4 pieces. Now
the problem is easy to solve!
• Subtraction can be shown in a similar way, as in “1⁄2 – 1⁄6.”
Again, a common denominator is needed. Exchange the 1⁄2
fraction piece for three 1⁄6 pieces to solve the problem.
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Multiplying and Dividing Fractions
• On a blank transparency, write “1⁄2 x 1 = ___.” Ask, “What
is 1⁄2 of 1?” Encourage students to find a fraction piece to display the answer and complete the equation:
“1⁄2 x 1 = 1⁄2.” Then ask, “What is 1⁄2 of 1⁄2?” Write the problem on the transparency (1⁄2 x 1⁄2 = ___), and then
have students solve it by displaying a fraction piece that is half as large as the 1⁄2 piece. Help them write the
solution: “1⁄2 x 1⁄2 = 1⁄4.” Repeat with other examples.
x
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• When students are comfortable finding fractional parts of a
whole number and can solve simple multiplication problems
like the ones above, introduce the fraction arrays. Begin by
reviewing a simple problem, such as “What is 1⁄2 of 1⁄3?” Guide
students to determine the answer mentally: 1⁄6. Next, write the
problem as an equation: “1⁄2 x 1⁄3 = ____.” Model how to use
fraction arrays to find the solution: Find the array showing 1⁄2
shaded blue and the array showing 1⁄3 shaded yellow. Place
the blue array on top of the yellow array so that the colored
arrays are perpendicular to one another. The resulting green
portion illustrates that 1⁄2 of 1⁄3 is 1⁄6.
• On a blank transparency, write “1 ÷ 1⁄3 = ___.” Ask, “How
many pieces will we get if we divide a whole into thirds?”
Encourage students to use the fraction pieces to solve the
problem: “1 ÷ 1⁄3 = 3.” Then ask, “How many pieces will we
get if we divide 2⁄3 by 1⁄3? In other words, how many 1⁄3 fraction
pieces are in a piece that equals 2⁄3?” Write the problem on
the transparency (2⁄3 ÷ 1⁄3 = ___), and then have students solve
it using the fraction pieces: “2⁄3 ÷ 1⁄3 = 2.” Repeat with other
examples.
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Probability
Materials: Spinner
Use the overhead marker to draw a line on the spinner, dividing it in half. Label the halves “A” and “B.” Ask
students, “Where is the spinner more likely to point when it stops—A or B?”
Discuss students’ predictions, and then ask a volunteer to spin the spinner and
Experiment Outcome
write the result on a chart, as shown. Ask, “What would happen if we repeated
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B
the experiment? Would it point to B again?” Continue spinning and recording the
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A
outcomes. Tally the results after 10 spins. Which outcome occurred more often?
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A
What would happen if you continued for 100 spins? Continue, stopping periodi4
B
cally to tally the results. Help children determine that, because the spinner is divid5
A
ed into 2 equal parts, both outcomes are equally likely.
Explain that the probability, or likelihood, of spinning an A is ⁄2, and the probability of spinning a B is also 1⁄2. The numerator of the fraction (1), represents the
number of outcomes that are being measured. The denominator (2), represents
the total possible number of outcomes. (You may also want to explain that probability may be expressed as a percentage instead of a fraction. In this case, the
probability of spinning an A would be 50%.)
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B
B
B
A
A
Repeat the exercise, dividing the spinner into 4 equal parts labeled “A,” “B,” “C,” and “D.” What is the
probability that the spinner will point to A now? Help students discover that it is 1⁄4, or 25%. What is the
probability for each of the other letters? (B = 1⁄4, C = 1⁄4, D = 1⁄4) Divide the spinner into 8 equal parts. What is
the probability that the spinner will point to any particular letter now? (1⁄8, or 12.5%)
Describing and Comparing 2-Dimensional Shapes
Materials: 2-D shapes
Display the shapes and help students identify their unique characteristics:
• A triangle has 3 sides and 3 vertices.
• There are some special kinds of triangles:
An isosceles triangle has two sides of equal length.
An equilateral triangle has three sides of equal length. Its angles are all equal (each one is 60˚).
A right triangle has one right angle (90˚).
In an acute triangle, all of the angles are less than 90˚.
In a scalene triangle, all sides are different lengths and all angles have different measurements.
• A quadrilateral has 4 sides and 4 vertices.
• There are some special kinds of quadrilaterals:
A trapezoid is a quadrilateral with one pair of parallel sides.
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel and equal in length.
• There are some special kinds of parallelograms:
A rhombus is a parallelogram in which all of the sides are the same length.
A rectangle is a parallelogram in which all of the angles are right angles (90˚).
A square is a parallelogram in which all of the sides are the same length and all of the angles are right
angles (90˚).
• A pentagon has 5 sides and 5 vertices.
• A hexagon has 6 sides and 6 vertices.
• An octagon has 8 sides and 8 vertices.
• A circle is a closed curved line consisting of points that are all the same distance from the center. A circle
has no vertices.
• An ellipse is a closed curve that looks like a narrow or flattened circle. An ellipse has no vertices.
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Encourage students to compare the shapes. Ask questions such as, “Which shapes have sides that are all equal
in length? Which one has the most vertices?” Then, play a “Secret Shape” game. Give clues such as, “The
secret shape has no right angles,” or “The secret shape has parallel sides.” Have volunteers remove shapes
after each clue until only the secret shape remains.
Describing and Comparing 3-Dimensional Shapes
Materials: 3-D solid shapes
Review the vocabulary as you point out the characteristics of 3-dimensional shapes:
• A triangular prism has two triangular bases and three rectangular faces.
• A cylinder has two circular, parallel bases that are the same size.
• A cone has a circular base and a curved surface that comes to a point.
• A square pyramid has a square base and four triangular faces.
• A cube has 6 square faces.
• A sphere has a curved surface with no edges or faces. It has a point exactly in the center that is the same
distance from any point on the surface.
Nets
Materials: Nets, 3-D solid shapes
Nets are used to study the attributes of solid figures, such as
the shapes that make up their faces. Display the nets one at a
time and compare them to the 3-D solid shapes. Ask, “Do you
think we could fold this net to form one of the 3-D solid
shapes? Which shape would it form?” Help students figure out
what shape each net represents. Prompt them to consider the
the number and shapes of the faces shown by a net. Which
faces are adjacent (next to) each other?
Area and Perimeter
Materials: Perimeter and area graphing board, 2-D shapes
(Note: The perimeter and area board is a 9" x 6" grid. Each square inch is subdivided into quarters.)
Exploring Perimeter
Explain that perimeter is the distance around a figure. Place the
large square on the perimeter and area board and model how to
use the grid to measure the length of each side. Remind students
that the perimeter is the sum of the length of all the sides of a
figure. Prompt them to find the perimeter of the square by
adding the measurements of all of the sides. Repeat with other
geometric shapes.
Exploring Area
Explain that area is the measurement of the space inside a
shape. Outline a 4" x 2" rectangle on the perimeter and area
board and help children count the squares to determine that
the area of the rectangle is 8 square inches. Next, place one of the geometric shapes on the grid and challenge children to determine its area.
Explain that another way to find the area of a geometric shape is to use a formula. For rectangles and
squares, the formula is “area = length x width.” For triangles, the formula is “area = 1⁄2 (base x height).”
Help children calculate areas for the square, rectangle, and right triangle. Then, use the perimeter and area
board to determine the areas of the same shapes and compare the results.
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Protractor
Materials: protractor, 2-D shapes
Draw an angle on a blank transparency and explain that an angle is
created by two rays that have the same endpoint. Demonstrate
how to use the protractor to measure the angle you drew. Then,
invite students to take turns drawing and measuring angles. Using
the 2-D shapes as a reference, challenge students to figure out
some basic facts about angles and triangles. For example:
• How many degrees are in a right angle? (90˚)
• What is the sum of all of the angles of any triangle? (180˚)
• What is the measure of each angle of an equilateral triangle? (60˚)
• What is the angle represented by a straight line? (180˚)
Decimal Boards
Materials: Decimal boards
Discuss the meaning of percent. Explain that it means “per hundred” or “for each hundred.” Point out that the
decimal board has 100 squares altogether. Then, use a wet-erase marker to shade 3 squares on the decimal
grid. Show students that 3 squares out of the 100 squares on the grid would be 3 per hundred, or 3%. Next,
explain that we can show this same quantity as a fraction: 3 out of 100 squares would be “three-hundredths,”
or 3⁄100. Remind students that we can also write this fraction as a decimal: 0.03. So, we can show the same number three ways: 3% = 3⁄100 = 0.03. Next, shade several rows. Help students figure out what percent is colored
and write the number as a percent, a fraction of 100, and a decimal numeral. Repeat with other examples.
Once students understand the concept, write a percent and ask a volunteer to shade the corresponding area on
the decimal board. Help students determine the equivalent decimal numeral and the equivalent fraction.
Plotting Ordered Pairs and Functions
Materials: Coordinate plane
Point out the horizontal and vertical lines on the grid and explain
that the horizontal line is called the x-axis because it is used to represent values of x. The vertical line is called the y-axis and is used to
represent values of y.
Remind students that an ordered pair of numbers, such as (5, 7),
represents a specific point on a coordinate grid. The first number, 5,
represents the value on the horizontal x-axis. The second number, 7,
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represents the value on the vertical y-axis. Demonstrate how to plot
the point. Repeat with other ordered pairs. Reverse the activity by
marking a point on the grid. Help students identify the point as an
ordered pair. For additional practice with plotting ordered pairs,
draw figures on the plane and challenge students to give the coordinates of each point on the figure.
Conversely, give students a set of ordered pairs to plot. What shape did they create?
To introduce how to graph a function, write a simple equation on the grid, such as “x + 2 = y.” Then, model
how to mark points on the graph for the equation. Say, “If x is 1, what is the value of y?” When students
respond that y equals 3, plot a point on the graph at (1, 3). Then ask, “If x is 2, what is y?” After students
respond, plot another point at (2, 4). Continue to calculate y for various values of x. Help children determine
that the points define a line, and extend the line all the way to (8, 10). Ask students if they can see a pattern
between x and y.
Graph points for “x + 4 = y” and compare the results. How are the lines for the two graphs similar or different?
Graph and compare other equations such as “2x = y” or “3x + 2 = y.” When students are ready, extend the
lessons to include negative numbers.
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Meeting Individual Needs
ELL
Before introducing an activity to the whole group, gather a small group and show them the manipulatives.
Identify each manipulative and have students repeat the word after you. Model how to use each manipulative. Have students record the names of the manipulatives in their vocabulary journals and explain how it is
used in their own words. Provide students with their own sets of manipulatives to use at their desks.
Reteach/Extra Support
Before presenting a concept to the whole group, meet with students who need extra support individually or
in a small group. Pre-teach the concept, and allow students to explore the manipulatives hands on. Giving
students the opportunity to have a tactile experience with the manipulatives will help them to focus their
attention when the concept is presented to the whole group.
Challenge
When students use manipulatives to solve math problems at their desks, invite volunteers to use the overhead projector and manipulatives to demonstrate how they solved the problem.
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