A Pattern-Block Toss Experiment - Everyday Math

A Pattern-Block Toss Experiment



Objective To guide children as they collect, tabulate, and interpret experimental data.

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ePresentations

eToolkit

Algorithms Practice

EM Facts Workshop Game™

Teaching the Lesson Key Concepts and Skills • Collect and organize data in a tally chart.  [Data and Chance Goal 1]

• Use probability terms to describe the likelihood of an event.  [Data and Chance Goal 3]

• Predict the outcome of a probability experiment and conduct a probability experiment.  [Data and Chance Goal 4]

Key Activities Children perform an experiment in which they determine the likelihood that a block will land on an edge when tossed. They determine whether doubling the thickness of a block changes the likelihood.

Family Letters

Assessment Management

Common Core State Standards

Ongoing Learning & Practice

Curriculum Focal Points

Interactive Teacher’s Lesson Guide

Differentiation Options ENRICHMENT

Displaying Shoe Lengths on a Line Plot

Predicting the Results of Rolling 2 Dice

Math Journal 1, p. 64 Student Reference Book, pp. 89A and 89B stick-on notes Children create a class line plot for shoe lengths.

Math Masters, p. 414 (See Advance Preparation.) per partnership: 2 dice, paper  tape  colored markers Children predict the results of rolling two dice and then test their predictions.

Math Boxes 3 5 

Math Journal 1, p. 66 Children practice and maintain skills through Math Box problems.

EXTRA PRACTICE

Using Probability Terms Children read Probably Pistachio and discuss probability.

Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 5.  [Patterns, Functions, and Algebra Goal 4]

Home Link 3 5 

Materials Math Masters, p. 70 Home Link 3 4 pattern blocks: triangles, squares, and trapezoids  tape  slate  Class Data Pad (optional)

Math Masters, p. 69 Children practice and maintain skills through Home Link activities.

Advance Preparation For Part 1, tape two square pattern blocks together to form a double-thick block; do the same with two triangle and two trapezoid pattern blocks. For the Line Plot activity in Part 2, draw a line plot on the board and add stick-on notes as shown in Part 2. For the optional Enrichment activity in Part 3, make 10 copies of Math Masters, page 414. Tape them together to form a grid scroll. Number the rows 2 through 12. Display this scroll on the board or a wall where children can reach it. For the optional Extra Practice activity in Part 3, obtain a copy of Probably Pistachio by Stuart J. Murphy (HarperCollins Publisher, 2001).

Teacher’s Reference Manual, Grades 1–3 pp. 118–128

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Unit 3

Linear Measures and Area

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Getting Started Mental Math and Reflexes

Math Message

On your slate, write sure, not sure, or impossible for each event.

Suppose you toss three pattern blocks into the air—a triangle , a square , and a . Which one has the best chance of landing trapezoid on one of its edges?

Tomorrow will be cold at the North Pole. sure Tomorrow it will be cloudy outside your school. Not sure A dolphin will sit in your chair tomorrow. impossible Scientists will find water on Mars. Not sure If there are 2 white socks and 2 black socks in a drawer and you close your eyes and pick 3, you will pick 2 socks of the same color. sure

Home Link 3 4 Follow-Up 

Briefly go over the answers. Ask children if the areas of the polygons in Exercise 1 are the same. no Have a few children explain their thinking.

1 Teaching the Lesson

 Math Message Follow-Up

WHOLE-CLASS ACTIVITY

Single blocks

NOTE The authors of Everyday Mathematics feel it is important to use correct terminology. Sometimes, however, it is necessary to simplify language to prevent confusion. For this reason, the authors choose to refer to the narrow side (or narrow face) of a pattern block in this lesson as an edge.

Have children vote for the shape they believe is most likely to land on an edge. Tally their votes. Save the tally chart to compare to children’s experimental results. Result

Double-thick blocks; the 6 blocks used in the experiment

Most likely to land on an edge

Teaching Master Name LESSON

35 䉬

Tally children’s guesses.

Date

Group Tally Chart

Block: Result

 Performing a Pattern-Block

Time

SMALL-GROUP ACTIVITY

Toss Experiment

As part of the experiment, children also toss a double-thick block of each shape. They determine whether doubling the thickness changes the chance of the block landing on an edge. Show children the six blocks that will be used in the experiment: the three single blocks and the three taped, double-thick blocks. Ask children to help you plan this experiment. In order to compare results easily, each of the six blocks should be tossed the same number of times.

Total

not on an edge Total number of tosses

(Math Masters, p. 70)

Children conduct an experiment to determine which of three pattern-block shapes is most likely to land on an edge (narrow side) after being tossed into the air. They toss a block of each shape a sufficient number of times (at least 50) to obtain reliable results.

Tallies

on an edge

Name LESSON

35 䉬

Date

Time

Group Tally Chart

Block: Result

Tallies

Total

on an edge not on an edge Total number of tosses

Math Masters, p. 70

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Block:

double □

Result Tallies Total on an edge ////\ ////\ ////\ ////\ / 21 not on an edge

////\ ////\ ////\ ////\ ////\ ////

Total number of tosses Sample group tally sheet for a double-thick square block

Block

Number of times landed on an edge 15 3 6

double

36

double

21

double

16

29 50

●

How many times should we toss each block? At least 50 times is best. Point out that by increasing the number of tosses, they are increasing the reliability of the results.

●

How will you divide the work? One possibility: Divide the class into six groups and assign one of the blocks to each group. For example, assign the single triangle to group 1, the double triangle to group 2, and so on. Then decide how many times each group member should toss the block to get a total of 50 or more tosses per block.

When the class has a plan, distribute the blocks and agree on how to toss a block. (For best results, all children should use the same technique.) A good way is to shake the block in cupped hands and release it about 2 feet above a surface. A dense carpet is best. Blotters will also reduce sound and bounce. Each group fills in the type of block they have on the top line of the tally chart on Math Masters, page 70. Then, they keep track of the number of tosses and the number of times the block lands on one of its edges. Children count their tallies to obtain group totals. Compile the results in a table on the board or Class Data Pad. Make sure that each group reports the same total number of tosses. Use language such as “The double-thick square block landed on an edge 21 times out of 50 tosses.”

 Discussing the Experimental Results

PROBLEM PRO P RO R OB BLE BL LE L LEM EM SO S SOLVING OL O LV VIN IIN NG

Have children rank the three single-block shapes according to the number of times that each one landed on an edge. ●

How well did you predict this ranking?

Review the tally chart with class responses to the Math Message. Results may vary greatly from class to class. It is likely that the single triangle will land on an edge more often than the single square or the single trapezoid. ●

Does doubling the thickness of a block change its chance of landing on an edge? Yes, each double block has a better chance of landing on an edge than a single block of the same shape.

●

Why should doubling the thickness of a block give it a better chance of landing on an edge? Sample answer: When you put two blocks together, you get twice as much edge part (area), but the pattern-block parts (areas) stay just the same. The extra edge part gives a double block a better chance of landing on an edge.

Sample results for one class; each block was tossed 50 times.

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WHOLE-CLASS DISCUSSION

Unit 3 Linear Measures and Area

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2 Ongoing Learning & Practice

 Displaying Shoe Lengths on a Line Plot

WHOLE-CLASS ACTIVITY

Number of Children

PROBLEM PRO P RO R OB BLE BL L LE LEM EM SO S SOLVING OL O LV VING VIN ING

(Math Journal 1, p. 64; Student Reference Book, pp. 89A and 89B)

Use pages 89A and 89B in the Student Reference Book to review line plots. On a stick-on note, have children record their shoe length measures from journal page 64. Have children share their measures while you record them on the board. Ask them to identify the shortest (minimum) and longest (maximum) lengths. Draw a line plot on the board or class data pad and label the axes “Number of Children” and “Shoe Length in Inches.” Write the horizontal scale in _12 -inch increments beginning with the shortest shoe length and ending with the longest. Ask children why the line plot can begin with a number other than zero. Sample answers: No one has a shoe that is zero inches long; no one in the class has a shoe shorter than the shortest shoe length. Invite children to place their stick-on notes above the number on the line plot corresponding to their shoe length. Have the class calculate the difference between the minimum and maximum shoe length (the range) and identify the shoe length that occurs the most (the mode). Next have children find the middle (median) class shoe length by removing one stick-on note from each end of the line plot one after another until only one or two remain.

9 12

9

10 12

10

11

Class Shoe Lengths (Inches) Line plot with stick-on notes

Number of Children

9 12

9

10 12

10

11

Class Shoe Lengths (Inches) The median is 9 1/2 inches.

Links to the Future Finding the maximum, minimum, mode, range, and median of a data set is a Grade 3 Goal. To provide practice with this skill, exercises such as this will be repeated several times throughout the year.

Student Page Date

Time

LESSON

35 

Math Boxes

1 __ 2 inch. Fill in the oval next to the best answer.

1. Measure to the nearest

 Math Boxes 3 5

2.

What is the perimeter? 3 cm

INDEPENDENT ACTIVITY



1 in.

(Math Journal 1, p. 66)

4 cm

2 cm

2 cm

1 1_ 2 in.

2 in.

Mixed Practice Math Boxes in this lesson are linked with Math Boxes in Lessons 3-7 and 3-9. Problem 6 previews Unit 4 content.

Ongoing Assessment: Recognizing Student Achievement

Math Boxes Problem 5



4 cm

1 2_ 2 in.

150

(unit)

3. Write , or =. Use a tape

4.

measure to help.

Add. Show your work.

> 16 inches 3 feet < 2 yards = 60 inches 5 feet 55 inches > 1 yard

Sample answer:

550 + 200 = 750 555 + 192

747 57–59 192

13, 146



20

Solve. 3×0=

14

0

=3+7+8+2

0×7=

29

9×0=

3 + 15 + 7 + 4 =

[Patterns, Functions, and Algebra Goal 4]

6.

Unit

9+1+4=

Unit

Ballpark estimate:

1 1_ 2 feet

5. Solve.

Use Math Boxes, Problem 5 to assess children’s ability to apply and describe the Commutative and Associative Properties of Addition. Children are making adequate progress if they are able to complete the 3- and 4-addend problems correctly. Some children may be able to combine the addends to make easier numbers.

3.5 cm

18.5 cm

143 144

0

=5×0

0 0

50 51

56

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Home Link Master Name

Date

 Home Link 3 5

Time



Describing Data

HOME LINK

35 Family Note

You can find information about minimum, maximum, range, median, and mode for a set of data on pages 79–82 in the Student Reference Book.

Children in the Science Club collected pill bugs. The tally chart shows how many they collected. Use the data from the tally chart to complete a line plot.

X X X X X X X X

Number of Collectors

0 1 2 3 4 5 6

Number of Children

/// ////\

0

// //

1

3 4 5 2 Number of Pill Bugs

1. What is the maximum (greatest) number of pill bugs found? 2. What is the minimum (least) number of pill bugs found?

4. What is the median for the data? 5. What is the mode for the data?

4 pill bugs 3 pill bugs 3 pill bugs

Home Connection Children create a line plot and use the data to find the maximum, minimum, range, median, and mode.

X X X X

Use the data to answer the questions.

3. What is the range for the data?

PROBLEM PRO P RO R OBL BLE B L LE LEM EM SOLVING SO S OL O LV LV VING VI VIN IIN NG

79–82

Please return this Home Link to school tomorrow.

Number of Pill Bugs

(Math Masters, p. 69)

INDEPENDENT ACTIVITY

2

6

6

pill bugs

3 Differentiation Options

pill bugs

ENRICHMENT

Practice

Unit

Make ballpark estimates. Solve on the back of this paper. Show your work.

 Predicting the Results

6. 67

28

95; 70

30

100

of Rolling 2 Dice

7. 33

29

62; 30

30

60

(Math Masters, p. 414)

Math Masters, p. 69

PARTNER ACTIVITY 15–30 Min

To further explore the concept of probability, have children predict the results of rolling 2 dice. Post the scroll described in Advance Preparation on page 194. Children work in partnerships. Each partnership needs two dice and a sheet of paper. First, ask children to predict which total from 2 to 12 will come up most often when two dice are rolled. Ask them to make one prediction per partnership. They record their predictions on a sheet of paper. Next, children draw a frequency table on the same sheet of paper and record the results of rolling the dice 30 times. As children finish their 30 rolls, have them come up to the scroll and make Xs with a marker to record the number of rolls for each sum.

Teaching Aid Master Name

Date

Time

Grid

The class graph should look something like a bell curve, although individual results may not. With enough rolls, 7 is usually the most frequent sum. Children discuss their predictions and individual results, and then compare them to the class graph. NOTE There are 36 possible combinations of two dice. Six of these have 7 as the sum: 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, and 6 + 1. Five combinations have 6 as the sum, five have 8 as the sum, and so on. So 7 is the most likely sum.

Children may continue and predict the results of rolling one die. Repeat the above experiment for one die. The scroll should be numbered from 1 to 6. Discuss why and how the results are different from those for two dice. Each number on the single die has an equal chance of being rolled, so the bars of the graph should be about the same length.

Math Masters, p. 414

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EXTRA PRACTICE

 Using Probability Terms

SMALL-GROUP ACTIVITY 5–15 Min

Read Probably Pistachio by Stuart J. Murphy in class or have children read the book themselves. Summary: Probability terms are used to describe events as Jack lives through an unlucky day. Children can use probability terms to describe events in their day.

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