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Catchy Title: Number Crunchers: monsters do the math
Theme/Topic of Lesson: number patterns, mathematical reasoning, addition and subtraction
Time Commitment: 3 one-hour lessons
Subject Area(s):
Mathematics - Patterns
Mathematics - Functions
Mathematics - Arithmetic
Grade Level(s): 2,3
Standards Alignment:
Class Challenge Question: What is a number function and how does it work?
Overview:
By 2nd or 3rd grade, students are familiar with the practice of skip counting, but may not have made the connection between skip-counting patterns and strategies and what they know about addition and subtraction. In this activity, students will use their understanding of skip counting as a bridge to identifying patterns and functions based on addition and subtraction. They will also use mathematical reasoning to analyze those patterns and functions.
To practice identifying and creating functions, students will create a function monster ("Number Cruncher") which will perform a function of students' choice by flipping a card upside down to reveal the appropriate result on the opposite side. By feeding cards into each others' monsters, students will have the opportunity to identify and analyze several different numerical functions.
Students will also apply the same strategies to identifying and working with functions based on subtraction. In the process, they will understand and explain the relationship between addition and subtraction. At the same time, they will be practicing addition and subtraction skills while becoming more familiar with the concepts.
Finally, students will have the opportunity to engage in a writing activity to reinforce their understanding of functions and how they are identified.
Students should have prior knowledge of skip-counting, as well as addition and subtraction. The teacher should be comfortable teaching math concepts using a variety of strategies..not just one method.
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Learning Objectives:
The Students will:
Use skip-counting skills to recognize patterns based on addition and subtraction - Explain and apply strategies for solving problems using addition and subtraction
Use mathematical reasoning to analyze patterns and functions - Identify and create functions based on addition and subtraction
- Understand the relationship between addition and subtraction
- Represent numerical relationships using appropriate symbols
Practice adding and subtracting
Assessment
Student performance will be assessed through the Number Cruncher's assessment rubric (Handout#1). The following activities will be assessed: Following the Rules, Making Monsters, Outsmarting the Number crunchers, Tracking Down the Monsters.
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Resources
Materials
Per classas determined by instructorPer student team/group of 2- empty shoebox
8 x 11 tagboard - eight 2 x 2 cards (blank) (These can be cut from tagboard.)
- tape
scissors - nametags (index cards) for monsters
Per Student
Vocabulary
- sequence - a list of numbers that go in a particular order based on a pattern
- function - a rule that shows how to get from one number to another number
- add - to skip forward from a lower number to a higher number (Note that this definition is specific to the context of the lesson.)
- subtract - to skip backwards from a higher number to a lower number (Note that this is specific to the context of the lesson.)
Procedures
Students will begin by reviewing skip-counting sequences that are familiar to them. They will then transfer the skills used in skip counting to analyzing number patterns based on addition and subtraction. They will use charts to record pairs of numbers and to see the relationships between the numbers in each pair. Next, students will create their own addition and subtraction "rules," or functions, and create cards that illustrate those functions with several examples. They will then build Number Cruncher monsters that flip the cards from one side to the other. (The initial number will be on one side of the card, and the resulting number on the other side.) These monsters will become tools with which students will practice analyzing and applying the various functions created by their classmates and performed by the monsters.
This lesson appeals to a variety of learning styles. Through knowledge construction, students will engage in mathematical reasoning and critical thinking, Artistically inclined students will enjoy the hands-on creation of monsters, and those who prefer writing will have an opportunity to bridge mathematics with writing.
During part of the lesson, students will work in pairs. It is recommended that students are paired by differentiation, specifically mathematics and artistic ability. That way, students will be able to learn from one another while also feel confident in their own abilities.
Students need to have previous familiarity with the concept of skip counting, and with basic addition and subtraction. (Addition and subtraction concepts and skills will be reinforced through this lesson.) For practice or informal assessment with skip counting, the teacher can refer to the online activities provided in the Resources section and engage students in one or more activities as appropriate.
Day 1: Identifying number patterns
Daily Challenge Question: How can you look at a group of numbers and describe the pattern using addition and subtraction?
one hour
Set-up Directions:
The teacher will need to provide a set of activity sheets for each student, and make sure each student has a pencil. Calculators can be distributed by student, pair, or small group, depending on availability and teacher discretion.
Teacher Presentation & Motivation:
Lead the class in a few familiar skip-counting sequences. Ask students how they know what number comes next in the sequence. (Choose a couple of volunteers to answer. At this point, you should accept a variety of explanations; it is not yet necessary for students to use the terms "add," "subtract" or other formal terms.) If they have simply memorized the sequences, choose one that is a little bit more challenging, or start with a higher number, so that students have to think about what comes next. If students are familiar with skip counting backwards, repeat the exercise with descending sequences. Explain to students that in this lesson, they will be using their skip counting skills to learn about patterns and functions, and to build Number Cruncher monsters.
Throughout this lesson, students will be building on previous knowledge as they are introduced to new concepts. Remind students frequently that they are using skills that they have already learned, but are simply applying those skills in new ways. At the same time, emphasize that looking for and identifying patterns is a skill in itself, and one that will be useful to them in solving many different kinds of problems.
In this lesson, vocabulary is introduced gradually and reinforced repeatedly. For example, students who are familiar with the concept of skipping will be encouraged to replace the term "skip" with "add," or "subtract" for "skipping backwards." (It is expected that students have already been introduced to addition and subtraction; however, this lesson provides another perspective on the concepts.) The teacher can reinforce students' understanding of terms by using the terms frequently in context and encouraging students to use the terms when they ask and answer questions.
Activity 1 - Finding patterns with addition and subtraction
In the introduction to the lesson, you reviewed skip counting with students. Now distribute calculators and show students how to generate sequences by adding the same number repeatedly. (For example, to add 5 repeatedly, enter a beginning number, then "+ 5," then "=." Continue pressing the equal sign to continue the sequence.) Ask students to describe the patterns they see during this exercise. For example, in a sequence based on adding 5, students will see the final digit alternate between two numbers. (e.g., 2, 7, 12, 17, 22, 27) Explore various sequences and discuss the patterns as a class. Point out the relationship between "skipping" and the "+" or adding operation on the calculator.
Now that students have reviewed skip counting and addition patterns, write a familiar sequence on the board. (For example, write "2, 4, 6, 8, . . . " ) Ask students what pattern they see or how they know what number comes next in the sequence. Solicit from students or point out that each number is equivalent to the previous number plus two. Reinforce this by going through the sequence and noting that 2+2=4, etc. Next, show students a sequence that illustrates skip counting by twos, but beginning with 1 instead of 2. (1, 3, 5, 7, . . . .) Emphasize that a sequence can begin with any number.
Provide a number out of order (e.g., 9) and ask students what would come next if they were skip-counting by twos. Repeat the exercise with different numbers until students seem comfortable with it. Ask them to explain how they figure out the answers.
Explain to students that you are now going to follow the same pattern, but this time you are going to organize the numbers into a chart. Make a chart with two columns on the board, overhead or chart paper (transparency a). Above the chart, write “Skip counting by twos.” (Choose another number if twos are too easy.) In the first column, write the number 1. Ask students what number would come next if they were skip counting by twos. (3) Write the number 3 in the second column. Explain that this is a skip-counting chart: you can put any number in the first column and they can figure out what is in the second column by adding (or skipping) two. Fill in the chart by placing several numbers in the first column and asking children to supply numbers for the second column.
Create a similar chart based on skipping/adding 3 (or another number). (As you go through this lesson, make sure students understand that adding is similar to skipping, and that the word "plus" indicates that we are skipping/adding. Use the term "adding" frequently throughout the lesson.)
Next, show students a chart that is already filled in (transparency b) and ask them what number is being added in each case. Write + 4 above the chart. Ask them how they figured it out. Repeat with transparency c, writing + 1 above the chart after students identify the rule.
Explain to students that they are using their understanding of skip counting to find patterns based on addition. Emphasize that what they are doing in this exercise is similar to what they have done before and that they can use similar calculation strategies.
Activity 2 - Following the rules
Explain to students that they will work independently to complete several charts similar to the one you have just completed together as a class. In each case, they will identify the “rule” on which the chart is based.
Distribute the Function Chart activity sheets (handout #1) and point out that in some cases, the rule is provided and students must apply the rule to fill in numbers in the two columns. In other cases, the columns are filled and students must identify the rule.
Ask students whether they understand the directions. If some seem uncertain, work through the first chart as a class before having students work independently on the others.
Circulate throughout the room to answer any individual questions that arise, and to make sure students understand the task. If some students finish early, provide them with blank charts (activity sheet a) and instruct them to create their own puzzles for each other. (For example, one child fills in the chart while another figures out the rule, or they create the rule and then fill in the chart together.)
When students have finished with the function charts, discuss their answers and strategies, and any patterns they noticed. (It is not necessary to go over every answer at this point, but enough to assess general understanding.) If some children have created their own charts, ask them about the rules they used.
Ask students to explain their strategies for filling in a number in the first column when only the second column was filled in. (Did they subtract, add, guess and check, use other techniques?) Talk about the last two charts from activity sheet 1 (based on subtraction). How did students figure out these rules? If they don’t spontaneously use the word subtraction, introduce the term.
Wrap Up:
Collect all activity sheets and other materials. Emphasize that students have used their skip-counting skills to find number patterns based on addition and subtraction. Explain that tomorrow they will create their own patterns and make Number Cruncher Monsters who will process numbers based on the rules they (the students) create.
Day 2: Building Number Crunchers
Daily Challenge Question: How can you make a machine that performs an addition or subtraction function of your choice?
one hour
Set-up Directions:
Make sure that each pair of students has a shoe box (students can be asked to bring these in ahead of time) with two holes punched in one side (not the top or bottom) about 3-4 inches apart. One hole should be at the bottom when the box is standing on one end, and the other should be a few inches above it. They will also need an 8 x 11 sheet of tagboard, eight blank cards (these can simply be 3 x 2 inch pieces of tagboard), a blank function chart sheet (Handout #1), a pair of scissors, tape, and a pencil.
The teacher may also want to create a sample Number Cruncher Monster so that students know what the end product will look like.
For decorating the monsters, students will need construction paper, markers or crayons, tape or glue, and any other art supplies you choose to provide.
Teacher Presentation & Motivation:
Review yesterday’s lesson by presenting the whole class with two charts on the board, overhead, or chart paper (transparencies d and e) and asking individual volunteers to fill in the columns according to the rules provided. Explain that today they will make monsters that crunch numbers based on their own rules.
Activity 1 - making monsters
Distribute materials to pairs of students. Explain that before making the Number Cruncher Monster, they will decide on the monster’s rule, or function. With their partners, they should choose a rule using addition or subtraction (e.g., + 4, - 3), then fill in the chart with 8 pairs of numbers based on their rule. Emphasize that the 8 pairs of numbers do not have to be consecutive; they may choose any number for column one, then use the rule to calculate the appropriate number for column two.
Circulate to assist students as needed while they fill out the charts. Take note of the variety of functions chosen. If several groups are choosing the same functions, encourage them to come up with more unusual rules for their monsters.
When the charts are complete, demonstrate to students how they will make their monsters. Using a pair of children’s scissors, they will enlarge the two holes previously poked in the side of the box. (These will be the holes through which cards are entered and removed.) These holes need to be only slightly wider than the cards and no more than half an inch to an inch high.
Next, students will use the tagboard to create a ramp inside the shoebox. The ramp should be positioned so that the lower end is attached to the same corner of the box as the lower hole. When a card is entered into the upper hole, it should hit the ramp and turn over as it slides down and comes out through the lower hole. The ramp can be secured by folding over an inch at the top and taping or gluing the resulting flap to the side of the box.
After demonstrating the process to the class, have students build their ramps in pairs. Once they have prepared the insides of the monsters, they can decorate the outsides as they wish. (You may wish to allow additional time, outside of the one-hour math period, for students to be creative in decorating their monsters.)
When the monsters are complete, have students write numbers on the cards based on the charts they have filled out. On each card, a number from the first column goes on one side, and the corresponding number from the second column goes on the other side. Make sure it is clear which side is the "top" and which side is the "bottom." (For example, this may be done through color coding.) Students can practice entering their cards in one hole and watching them come out the other hole with the opposite side up. They should make sure this works smoothly, and they can adjust the angles of their ramps as needed.
Wrap Up:
Point out to students that they have created a "function" machine: whatever number is entered, the monster adds or subtracts based on the rule the students have chosen. (If time allows, students can check out each other's monsters.) Explain that tomorrow they will explore each other's monsters and figure out the monsters' different rules.
Have students clean up all materials and place monsters together in a safe area. Make sure that each set of cards stays with the appropriate monster.
Day 3: Crunching numbers
Daily Challenge Question: Can you outsmart the Number Crunchers?
one hour
Set-up Directions:
Each pair of students will begin with their own monster and set of cards, five blank charts (activity sheet 1), and a pencil.
Teacher Presentation & Motivation:
Explain that the class now has many Number Crunchers, and that students will be able to use their understanding of addition and subtraction to figure out each monster's rule, or function. Write the word "function" on the board and explain that this word is used to describe what each monster does (e.g., adds 3 or subtracts 5). Explain that students should not tell each other what their monster's function is, because part of their job for today will be to figure that out for themselves. Before they begin, ask each pair of students to give their monster a name, and to write the monster's name on a name tag which can be placed beside the monster. Take a few minutes to allow students to introduce their monsters to the class.
Activity 1 - Outsmarting the Number Crunchers
Have students work in the same pairs in which they built their monsters. (However, each child will fill out his/her own charts.) Have them rotate around the room, visiting monsters with their blank charts. When visiting a monster, the students should first write the monster's name at the top of the page. Next, they will feed the monster its cards, and use their charts to record the "in" and "out" numbers of each card. Then they will record the monster's function at the top of the page. After a few minutes, ask students how many cards it took before they figured out the monster's function. (It may be only one or two.) Next, tell them that they should be able to predict the "out" number by looking at the "in" number and adding or subtracting according to the given function. Ask them to record the "in" number on their charts, along with their predictions for the "out" number. Then enter the cards to see whether their predictions were correct.
Have students rotate and repeat this exercise with four or five different monsters. Then lead a discussion about their strategies for identifying functions and for predicting outcomes based on the functions.
Activity 2 - Tracking down the monsters
At this point, each pair of students should be working with one monster. Ask each pair to write the monster's function on the monster's name tag. Next, collect all cards and mix them together in a box. Ask students how they will be able to return the cards to the appropriate monsters. Allow one or two volunteers to explain how this can be done. Next, distribute several cards to each pair of students and allow them to match their cards to the appropriate monsters. (To make this flow more smoothly, you might suggest that they identify all cards' functions first, then look for the matching monsters.)
When students have matched all their cards with the appropriate monsters, have them return to their seats and write a short explanation in their math journals (or a separate sheet of paper, if they don't have math journals) of how they matched the cards with the monsters. Collect these explanations for evaluation.
Wrap Up:
When the students have returned to their seats and written about their strategies, ask for volunteers to explain how they matched the cards to the monsters. During this discussion, reinforce the use of the terms "add," "subtact," and "function." Ask them to compare their strategies for adding and subtracting. How are these operations alike and different? Ask students how their charts helped them identify the monsters' rules.
Collect charts and have students put the monsters away, along with the corresponding cards. The monsters can be displayed in the classroom and used later for more complicated operations, as students build on their understanding of addition, subraction, (and other operations) and functions.
Enrichment Options
Field Experiences
Extension (Not necessarily field experience)
Mix together the cards from three (or more) monsters. Have students look at both sides of each card (either by hand or by feeding them to a monster), then place them in piles according to their functions. (i.e., each pile represents one function)
Ask students to work with one monster (whose function they know) and have them put the cards in backwards. (i.e., bottom side up) Ask them how they would describe the resulting function. Write on the board or overhead the original and "reverse" function for each monster. Ask students to comment on the pattern they see. Why does it work this way?
Cross-Curricular Extensions
Language arts
Have students write descriptions of their monsters and what they have learned about numerical functions. Also, they can compose fictional stories about their monsters.
Fine arts
Students can use artistic skills in decorating their monsters. They can also draw pictures of their monsters to illustrate their descriptions and/or stories.
Science
Students can discuss the mechanics of the slide (inclined plane) inside the monster. How could you design monsters differently to make cards come out right-side-up, or flip over more than once? This can be expanded into a lesson on simple machines, since a ramp or inclined plane is a kind of simple machine.
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As a reflective practitioner, note how this lesson could be adjusted after its initial implementation. How successful were the students? What did the assessment demonstrate about the students' learning? What skills do the students need to revisit? What instructional strategies worked and what made them successful? What will you change the next time you use this lesson? Why?
Author: Amy Stuart
Modified by: Donna Schnupp
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