National Council of Teachers of Mathematics Standards for School Mathematics

“The Standards for school mathematics describe the mathematical understanding, knowledge, and skills that students should acquire from prekindergarten through grade 12. Each Standard consists of two to four specific goals that apply across all the grades. For the five Content Standards, each goal encompasses as many as seven specific expectations for the four grade bands considered in Principles and Standards: prekindergarten through grade 2, grades 3–5, grades 6–8, and grades 9–12. For each of the five Process Standards, the goals are described through examples that demonstrate what the Standard should look like in a grade band and what the teacher's role should be in achieving the Standard. Although each of these Standards applies to all grades, the relative emphasis on particular Standards will vary across the grade bands.”

The following are a compilation of different activities that cover each of the ten standards from a constructivist perspective, in which the students construct their own understanding of mathematical concepts.  Although there is an activity for each of the standards, they are not mutually exclusive; many of the activities relate to more than one of the standards.

The Core Curriculum Content Standards

There are 10 different content standards for math.  Please feel free to use the following links to jump right to a specific standard under which you will find a lesson plan that relies on constructivist principles and utilizes different sensory input so as to reach as many students as possible:

Standard #1: Numbers and Operations

Name of Activity: Salute

Objectives:

1. Given a deck of cards with the picture suits removed and knowing one addend and the sum of an addition sentence, students will be able to identify and say the missing number in the addition sentence.
2. Working in groups of three, students will be able to work cooperatively with their peers to compute addition and subtraction of numbers up to 10.

Materials: one deck of cards per group of three students

Procedure:

1. Place students into groups of three students.
2. Give each group of three students a deck of cards with the picture cards removed.
3. Have two of the students sit facing on another (on opposite ends of a table or desk) with the third student sitting so they can see the other two students (like a triangle).
4. The two students who are facing each other are the “Guessers” – they both choose a card from the deck without looking at it.
5. The third student says “SALUTE” and the two “Guessers” hold their card up to their forehead so that the “Guesser” sitting opposite from them can see the card’s value.  The “Guessers” should not know the value of their own card.
6. Now the two “Guessers” can see each other’s card but not their own.  The third person can see the value of both of the cards and tells the two “Guessers” the sum of the two cards.
7. Knowing the sum of the two cards, the “Guessers” should be able to figure out the value of their own card by using subtraction.

Rationale:

According to the NCTM (www.NCTM.org), “numbers pervade all areas of mathematics. The other four Content Standards as well as all five Process Standards are grounded in numbers.  Central to the Number and Operations Standard is the development of number sense…[students] solve problems using the relationships among operations.”  This activity revolves around the concept of solving problems using the relationships among operations.  More specifically the relationship between, and reversibility of, addition and subtraction are addressed.  In this activity, students are made aware of the sum of the two cards, and they must mentally compute what the missing part of the addition sentence is by subtracting the value of the other “Guesser’s” card from the sum.  This requires that students have knowledge of the rules regarding addition and subtraction, and have a satisfactory sense of numbers and their properties.

Standard #2: Algebra

Name of Activity: Math Scramble

Objective:

Given a card with a number written on it, students will use math skills and teamwork to solve basic algebraic math problems.

Materials: Index cards with a number 0-9 on each, several index cards with plus, minus,

division, and multiplication signs
Procedure:

1. Each student should be given a card with a number on it from 0 - 9.
2. The teacher calls out a number, which is the answer to either an addition, subtraction, multiplication, or division problem.
3. Students need to use the number on their card in conjunction with the numbers on other student’s cards to figure out a mathematical equation.  Plus, minus, division and multiplication signs should be in the front of the room - students should get the signs they need to answer the problem.
4. Students arrange themselves in order to form a correct problem.
5. The teacher selects the problem that was the most difficult to arrange, or the one that uses the most students and each of those students gets a point.
6. After a few rounds, students can exchange their numbers.
7. The student with the most points at the end of the game gets a prize.

Rationale:

According to the NCTM, algebra encompasses the relationships among quantities and the mathematical study of change.  The activity requires that students begin to think about numbers and their relationships algebraically by mentally computing different possible combinations of numbers and operations.  This skill is a precursor to a more formalized study of algebra.  Math Scramble is an effective activity for students in the pre-algebra stage of mathematics because of the multiple unknowns that are contained within the activity.  Students have the opportunity, given the specific number written on their index card, to construct their own equations equaling a specified number using an indeterminate amount of other numbers and operation signs.  Having students think in terms of “unknowns” to solve mathematic problems is the beginning step in introducing the concept of variables.  According to the NCTM, “the word algebra is not commonly heard in elementary school classrooms, but the mathematical investigations and conversations of these students in these grades frequently include elements of algebraic reasoning.”  This activity is an example of one that includes elements of algebraic reasoning, providing fundamental skills that will act as the building blocks of future education.

Standard #3: Geometry

Name of Activity: Shape Pictures

Objective:

Given different paper shapes, students will be able to arrange the shapes to form pictures, and identify and count the different shapes that they used in their pictures.

Materials: sheets of paper with a variety of shapes on them, scissors, glue, construction

paper, pencil, and crayons or markers

Procedure:

1. Give each student several sheets of papers with many different shapes on them.
2. Have students look at the different shapes and cut them out.
3.  After all the shapes are cut out have the students put them together to make different objects or a whole picture scene.
4. Have the students keep track of how many of each type of shape they use. The students should make a chart of how many of each shape they used (they can decide what type of chart they will use and explain their chart to the class)
5. Prizes should be given to the students who use the largest amount of shapes, the largest variety of shapes, and who can explain their charts the most thoroughly.

Rationale:

The NCTM standards suggest that geometry and spatial sense are “fundamental components of mathematical learning.”  Furthermore, after being provided instruction concerning geometry, students should be enabled to analyze characteristics and properties of geometric shapes.  This activity addresses such issues as the characteristics of shapes, shape identification, and spatial relationships between shapes.  In Shape Pictures, students must identify and count all of the different geometric shapes that they have chosen to use to create a larger picture.  Additionally, this activity speaks to the concept of shape composition; that bigger pictures are made up of many smaller, simpler geometric shapes.  Through a creative, hands-on activity, the ideas of naming shapes based upon their characteristics and charting data are reinforced.  As the ability of students to understand and detect the relationships among shapes becomes more refined, students should be able to move toward more abstract concepts such as theorems and proofs.

Name of Activity: Measure It!!

Objective:

Given a ruler, pairs of students will be able to measure different items in the classroom in either inches or centimeters and record their results in a chart.

Materials: Rulers, paper, pencil

Procedure:

1. Ask your students to get a partner and give each group of two a ruler.
2. Give each pair of students a piece of paper on which they should make a list of items that they want to take turns measuring around the classroom, items that could be suggested include:
• a new pencil
• an eraser
• the door from the doorknob down to the floor
• a bookbag
• a notebook
3. Have the pairs of students move around the room and measure and record the length of the objects on their list
4. After the students have measured all of the items on the list have them put their results on a graph from shortest to tallest.
5. After the pairs of students are done constructing their graphs have them present their findings to the class
6. After the presentations have a whole group discussion comparing and contrasting the results that the pairs of students obtained

Rationale:

Measure It! addresses both of the instructional goals listed under the Measurement Standard including understanding the measurable attributes of objects and the application of appropriate techniques and tools to determine measurements.  This activity requires that students use measuring tools (in this case a ruler) to find the length of a variety of objects that are encountered in their classroom, reinforcing the NCTM statement that, “the study of measurement is crucial…because of its practicality and pervasiveness in so many aspects of every day life.”  By having the students choose the objects around the classroom that they would like to measure, they will be engaged in the activity from beginning to end because they will be interested to see what objects the other pairs of students chose to measure.  Measure It! also highlights to concept that mathematics exists all around us, not only during a math lesson involving numbers and operations.

Name of Activity: M&M Graphing and Probability

Objective:

Given a small bag of M&M’s, pairs of students will be able to count, sort, and classify M7M’s by color, record data on a chart, use the data from the chart to create a bar graph, analyze and interpret their data, and use their data to figure ratios and determine probability.

Materials: 1 small bag of M&M’s per pair of students, pencil, paper, rulers, crayons or

markers, teacher generated chart

Procedure:

1. Put students into pairs and give each pair one small bag of M&M’s
2. Instruct the pairs of students to open their bags of M&M’s (but not to eat any) and sort and classify the M&M’s according to color
3. Have the pairs of students to record their data on the teacher generated chart and tape their charts on the board
4. Have a whole group discussion about the similarities and differences between the charts
5. Put students into groups of 4-6 (with the original pairs of students in the same group) and have the students combine their charts to make a new chart illustrating the results
6. After discussing bar graphs, have the groups create and color a bar graph using the new data (the data recorded after combining the pairs of students)
7. For lower grades (1-3), analyze the data in terms of most, least, greater than and less that.  For higher grades (4-5), ask the students to determine the ratio of each color of M&M’s in a bag and predict the probability of selecting any one color at random from a bag
8. Eat and enjoy the M&M’s

Rationale

According to the NCTM, “to reason statistically, which is essential to being an informed citizen, employee, and consumer, students need to learn about data analysis and…probability.”  M&M Graphing and Probability addresses the student’s ability to collect, organize, and display data in order to answer a question, and also to develop predictions that are based on the data.  Most students will be engaged in this activity because at the end they all get a sweet reward, they get to eat their M&M’s!  Like many activities dealing with data and probability, this activity lends itself to use with students of varying ages and abilities depending on the questions that are asked and accommodations that are made.  For students in lower grades this activity can introduce basic vocabulary involved with data collection and probability, and for students in higher grades it can introduce and reinforce the rules regarding ratios and probability.

The Process Standards

Standard #6: Problem Solving

Name of Activity: Fun With Estimation

Objective:

Given a shopping list, a money limit, and specifically priced items, students will use estimation to keep a running total of items to be purchased and use problem solving strategies to figure out what items to purchase in order to avoid over spending.

Materials: empty packaging from the grocery store, timer, calculators

Procedure:

1.      Put students into groups of 4 and assign each member a number 1-4

2.      Call student 1 from each group and give each one an identical shopping list and a money limit

3.      Set your timer for 1-2 minutes

4.      Upon your signal, students will select their items from the store and purchase as many items as possible without going over their

money limit.  Allow students to take items back to their group so they can assist in estimating the total.  If the student has money left,

and time allows, they can go back to the store a purchase additional items

5.      When the timer rings, have the students return to their groups and add up the actual prices of the items they selected.  If they

estimated carefully and did not go over their budget their group gets a point.

6.      Have each group tell their total to the class.  The team group that came the closest to the money limit without going over receives a

bonus point.

7.      Begin the process again with group member 2 and continue until all group members have had an opportunity to participate.  Keep a

running total of each teams points – the team with the highest point value wins and earns a reward

Rationale

In this activity students must use estimation and problem-solving techniques to figure out how to buy a certain number of grocery items without spending more money than is allotted to them in their budget.  In groups of 4, students must work cooperatively to apply and adapt strategies to solve the problem and must also monitor their own problem solving.  This estimation activity allows students to “solidify and extend their knowledge…[by] engaging in a task for which the solution is not known in advance,” (www.NCTM.org).  Additionally, during this activity, students must work with other students in their group in order to develop a strategy for solving the problem and must reorganize and reflect upon that strategy throughout the course of the game.

Name of Activity: Sort It Out

Objective:

Given an assortment of objects or pictures, students will be able to use logical reasoning and attributes to sort objects into overlapping circles (Venn Diagrams)

Materials: magazines, newspapers, tape, scissors, string

Procedure:

1.      Set out an assortment of objects or pictures.  Use the string to make overlapping circles (Venn Diagram) on the front table.

2.      Secretly decide on a set of attributes for sorting the objects.

3.      Invite a student to put an object in one of the three areas, making a guess as to where it should go based on your secret attributes.

Say “yes” if the item was sorted correctly, or “no” if it was not (and return the item to the pile).  Repeat with the next student.

4.      Once students have sorted 3 or 4 items correctly according to the secret attributes, ask them if it is getting easier to sort the items.

Also ask what is making it easier to sort the items (try to get them to explain what is making it easier for them to sort the items).

5.      The game is over once the students have guessed the secret attributes and have sorted all of the objects.

6.      Break students up into groups of 4-5.  Assign one student to be the “Attribute Assigner”.  That person will develop secret attributes

by which the items should be sorted.  Repeat the game in the small groups.

Rationale:

In this activity students must use reasoning to figure out the attributes by which a variety of pictures have been sorted into a Venn Diagram.  Initially the activity is based on trial and error, but as is progresses patterns should emerge and become more apparent to the students.  As the attributes by which the pictures are placed into the Venn Diagram become more apparent to the students, they must reflect upon their own reasoning to answer critical thinking questions as posed by the teacher regarding why it is becoming easier to sort the pictures.  The NCTM suggests that in lower grades reasoning and proof are often inductive through patterns and specific cases, but as students move through the grades students learn to make deductive arguments using mathematical truths.  This activity is an example of using inductive reasoning to sort different pictures based upon different attributes and patterns of sorting.

Name of Activity: Who is the Teacher?

Objective:

Given two-step addition word problems, pairs of students will solve the problems in a “teacher-student” dyad by communicating the steps taken to solve the problems as they are completed, and correcting one another if problems arise

Materials: worksheet of two-step addition word problems, paper, pencil

Procedure:

1.      Put students into groups of two and assign one student to be the “teacher” and one student to be the “student”.  The “student’s” job

is to solve the problems and the “teacher’s” job is to prompt support the “student” and prompt him or her in the right direction if they

need help

2.      Have the “student” attempt the first word problem.  As they work through the problem they should be orally communicating to the

“teacher” what they are doing and why.

3.      The “teacher” should encourage the “student” if they are doing the problem correctly (or if the explanation of their steps is logical)

and should make suggestions to the “student” about how to solve the problem if questions arise.

4.      Have the students switch roles for every question until the worksheet is completed

Rationale:

According to the NCTM, when students communicate and “justify their reasoning to a classmate…[students] gain insights into their thinking.”  In this activity, as pairs of students attempt to solve word problems, they must communicate to one another how and why they are solving the problem.  In order for this activity to be successful students must “be able to organize and consolidate their mathematical thinking, communicate their mathematical thinking clearly and coherently to peers, analyze and evaluate the mathematical thinking of others, and use the language of mathematics to express mathematical ideas concisely,” (www.NCTM.org).  Who is the Teacher? requires students to act in two different roles, one in which they must communicate their mathematical ideas clearly to a peer and one in which they must analyze and evaluate  the mathematical thinking of a peer.  This game spans the gamut of instructional objectives related to the Communication Standard.

Name of Activity: What’s the Pattern?

Objective:

Given the recording of a piece of music, the students will be able to identify the musical patterns that are present and begin to understand how pieces of music are constructed and the different parts that make up a song

Integrated Subjects: Mathematics, Language Arts, Music

Materials: recording of a popular song, paper, musical instruments

Procedure:

1.      Play part of a popular song to the class. After the class has listened to the song once, the play it a second time and instruct the class

to listen carefully to find a pattern in the music.

2.      Review how patterns exist in Mathematics

3.      Give the students some examples of how different patterns exist in music and have them describe what they think the pattern/s may

be.

4.      Play some simple patterns on different instruments.  Give each student in the class an instrument and one by one have them copy

5.      Put students into pairs; make sure that each member of the pair has a different instrument.

6.      Have the students work together to produce a song using the instrument to make up a pattern and have them write lyrics that have to

do with school – give them 30 minutes to compose their songs

7.      Have each group come up to the front of the room to perform their song. The rest of the class will try to imitate the pattern with their

own instruments

Rationale:

What’s the Pattern? is an activity that deals with the connections that exist between mathematics, music, and language arts.  This is in line with the instructional objective under the Connections Standard that students should be able to “recognize and apply mathematics in contexts outside of mathematics,” (www.NCTM.org).  It is important for students to realize that mathematics does not exist in solitude.  Alerting students to the interplay between mathematics and other subject areas connects mathematical concepts to the every day lives of the students.  The majority of people enjoy music.  With that being assumed, connecting mathematical concepts to an enjoyable medium will enable students to be better adept in using insights gained in one context to verify and understand the same concepts in another context.

Standard #10: Representation

Name of Activity: Congruent Mobile

Objective:

Given different felt shapes, students will construct a mobile that demonstrates their understanding of congruency by including two pairs of congruent shapes and one pair of non-congruent (but similar) shapes

Materials: variety of felt shapes, hangers, different colored yearn, markers, hole puncher

Procedure:

1.      Review the concepts of congruent and similar on the chalkboard.

2.      Have the students choose three sets of felt shapes – two that are congruent and one that is similar

3.      Students should label the two pairs of congruent shapes as “congruent” and the pair of similar shapes as “similar”

4.      Have students choose the color yarn that they would like to use

5.      Have the students tie their pairs of labeled shapes to the hanger using yarn

6.      Hang the mobiles from the ceiling

Rationale:

The Congruent Mobile accomplishes the first instructional objective under the Representation Standard by enabling students to “create and use representations to organize, record, and communicate mathematical ideas,” (www.NCTM.org).  By selecting and labeling the pairs of shapes that are congruent and similar and then attaching those pairs to a hanger to be hung from the classroom’s ceiling, the students are using representations (the pairs of felt shapes) to communicate the concepts of congruent and similar shapes.  The Congruent Mobiles act as visual representations through which mathematical concepts are made clearer to the students who constructed them and also communicate the mathematical ideas related to congruency and similarity to others who will view them.  The goal of this activity is to enable the students to represent the concept of congruency in a way that makes sense to them.