# 2nd Grade Fractions Lesson Plan Equivalance

## Discussion/Introduction

In first grade our students were briefly introduced to fractions—or, more specifically, halves and quarters– as equal divisions done on rectangles and circles. They identified halves and quarters and divided their own samples into two or four equal parts. The emphasis was on neat, symmetrical divisions that made it easy to check equivalence.

In second grade, though, we get to work with the concept of strange looking equivalences— portions which are equal but which are different shapes. This concept will probably be counterintuitive to many of your students, but it is an important concept nonetheless: a crucial part of their understanding of space, area, and sameness.

This lesson plan gives a brief review of halves and quarters, introduces the new kid on the block—thirds—and goes on to discuss strange looking equivalences and what it means for portions or fractions to be the same.

## Objective

That students would gain familiarity in partitioning circles and rectangles in two, three, and four equal shares, and would understand that equal shares of  identical wholes need not have the same shape. (Common Core 2.G3)

## Supplies

• One orange
• Construction paper, pre-cut into a quantity of identical rectangles and circles: 3 circles and 5 rectangles, preferably in an assortment of different colors
• Markers
• Scissors
• Scotch Tape
• Beads or other small math manipulatives (12 per student)

## Methodology/Procedure

Start class by reviewing fractions, as learned in first grade. Tell your students that you like eating orange every morning, but you only have one orange for today and tomorrow; ask them how much orange you should eat today (half—one of two equal parts). Ask them how you should divide it in half (across the middle), and ask them divide their first circle by drawing a line. Then ask them how much orange you can eat today if you need to make it last for four days. (one fourth – one of four equal parts). Ask them to draw this on their second construction paper circle cutout.

Take the orange out of your desk and tell them that actually, you’ll be able to buy a new orange to eat on the fourth morning. It’s just three days that this orange has to last. Ask how much orange you can eat today.

Divide the orange into three equal portions, and tell the class that they are called thirds. Tell them that when you are making thirds out of a circle there’s no middle line to divide on. Show them, or draw on the backboard, a picture of an orange cross section cut into thirds; demonstrate how you can start by making a line through the middle, to the center point, and then continuing it as a (wide mouth) Y.

Ask your students to draw lines and divide their last construction paper circles into thirds. Have them cut out the segments and lay them over each other to check their own work.

Now tell them to take out their first rectangle, and tell them you’d like this to be divided into thirds. Ask them how they would go about it.

After they have had plenty of time to experiment call up any students who have been especially successful to show their techniques. If they are all still fumbling, show them your rectangle, creased to show thirds. Give them a chance to imitate this with their own paper, and have the first successful student show them how folding the two sides over the middle segment and creasing the fold will give three equal sections— three thirds.

Ask which is larger, a half or a third. And which is smaller, a fourth or a third.

Give each student a pile of twelve beads, tiles, or other math manipulatives, and ask them to divide it into thirds. Let them try to figure this out by themselves before you give them any input. You can continue this exercise to halves and quarters.

When your students feel comfortable making divisions in halves, thirds, and quarters, it’s time to go on to strange looking equivalences. If your students had forgotten some of their first grade work, you may want to simply spend the rest of the class period playing with fractions and save this second half of the lesson till your next class. If your students had no difficulties with your preliminary exercises, though, you can go straight on.

Ask the students to divide their second rectangle in halves. Choose two students who have halves that are proportioned differently, and have them show their halves to the class. Ask which is larger.

If there is the slightest doubt in anyone’s mind, point out how, though one is wider, the other is taller. Demonstrate equivalence by cutting and taping the rectangles till they become the same size and shape. Only after you have demonstrated it tell them that half of an object is always the same size as any other half of the same object, no matter what shape it takes. You don’t want them to accept this as a rule for how halves should behave, but simply as a demonstrated fact.

Provide each student with three more identical rectangles, pre-cut out of three different colors of construction paper. Ask the students to draw lines to partition each rectangle in fourths, and tell them you’d like the partitions to be different in each rectangle.

Give the students the attached worksheet and ask them to mark the areas that are equivalent.

## Common Core Standards

In 2.G.3, 2nd grade geometry item 3, the Common Core State Standards for Mathematics reads:

2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

## Web Resources/Further Exploration

You may also want to point your students to some of our other fractions resources and links:

# M&M Fun: A Bar Chart Lesson for 2nd Graders

Sample Bar Graph

## Discussion/Introduction

After weeks and weeks of arithmetic and mental math that tax my students’ minds to the utmost, I always enjoy getting to the graph section of our curriculum. It’s a breather, almost, and the change of pace is very welcome.  At the same time, our unit on graphing is a very important one—as students enjoy making colorful charts and graphs they learn how to condense real-world data into a mathematical format that is easy to understand, concise, and easy to analyze.

A 2nd grade bar chart lesson can be just complex enough to be exciting, but there’s no need to go into the complicated situations that will leave your students scratching their heads in confusion. The free chart-maker at Meta-Chart (http://www.meta-chart.com/) is a wonderfully easy way to make professional,  streamlined charts and graphs; all you do is plug in your data and—eureka!—out comes the graph.

## Objective

Students will learn how to analyze data on a simple single-unit bar chart with four categories. They will learn to solve simple put-together, take-apart, and compare problems using information presented in a bar graph by making use of a free online bar graph maker.

## Supplies

• A mug with around 20-30 colored M&Ms, and ~8 M&Ms for each child in the bag
• 1 sheet of graph paper for each child
• Markers
• Graph printouts from http://www.meta-chart.com/bar (or, if you have projector capabilities, meta-chart graphs prepared and saved on your laptop).

## Methodology/Procedure

When students are investigators what they learn becomes part of them. Bar graphs lend themselves well to a discovery way of teaching because they are, in a way, a special code that can be discovered. Your students will have had some exposure to graphs in grade 1; some will have even done very basic work with bar graphs. They have all the tools they need to figure out what the bar graph is telling them if you give them the time and encouragement to figure it out by themselves.

Teacher: Look, I have a mug here with M&Ms in it. You can’t look inside right now, but here is a graph that tells you what the colors of the M&Ms are.

[Give students a chance to look at it and think about what it is saying.]

Teacher: Can anyone tell me which color I have the most of?

Student: Red!

Teacher: Yes! You are correct. How did you know?

Student: The bar labeled red is the longest.

Teacher: You’re right. Can you tell me how many red M&Ms there are in my cup?

If a student gives the right answer, applaud him and ask for an explanation why. If no-one knows, ask some more leading questions.

Teacher: How long is the red M&M bar? Is there a number which tells how long it is?  What do you think that number might be telling us?

Once the students have figured out how many red M&Ms there are, reinforce that interpretative ability by asking questions about the other colored M&Ms in the mug. Then go on to problems related comparing.

Teacher: How many more red M&Ms do I have than blue?

Lead the students to discover that they can find the difference without doing subtraction by simply noticing how much further the red line sticks out.  Ask questions comparing each of the other lines.

When your students are comfortable comparing, go on to simple put-together and take-apart problems using the data on the graph.

Teacher: If I don’t like red or green M&Ms and decide to throw those ones away, how many will I throw away?

Given the opportunity to discuss and brainstorm, your students should have no trouble solving this. If their thinking was anything less than automatic, follow this up with a similar question. If your dog only can eat yellow or brown M&Ms, how many will he have?  Then ask a take-apart question:

If I actually like green M&Ms and want to have as many green M&Ms as red, how many more green M&Ms do I need to put in the mug?

Once the students are comfortable doing a variety of problems, have them each draw two axis on their own graph paper, and label the vertical axis with numbers 1-6. Distribute around eight M&Ms to each child, and have them draw a bar graph. They can color each bar the bars the same color as the M&Ms for easy labeling. Post the graphs at the front of the room, and discuss which child has the most red, the most green, the most yellow, and so on.

#### Evaluation

You’ll be able to tell how well the students understood the concept of a bar graph by how much help and hand-holding they need when it comes to drawing their own graphs.  Quick comparing and put-together questions on the smaller numbers of their own M&M collections and graphs will give you another way of testing their ability to understand the logic behind bar graphs.

###### A Single Unit Bar Chart Lesson in Your Classroom

If you use my lesson plan I’d love to hear from you regarding your experiences—did you have fun with your students? What is your preferred way of teaching  a 2nd grade bar chart lesson? Please comment!

## Common Core Standards

The Common Core recognizes the importance of learning how to use graphs for data representation and manipulation in the early grades. For second grade, under measurement and data  [ 2.MD.10 ], the Common Core State Standards for Mathematics reads ‘Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put- together, take-apart, and compare problems using information presented in a bar graph.’

## Web Resources/Further Exploration

The chart maker at http://www.meta-chart.com is a convenient, easy to use, free way of preparing charts or graphs for any of your classes!