In my opinion, the process of **repeated subtraction** is very important for students to practice. With repeated subtraction, we are actually asking this question: “How many _____ in _______?” If the problem was 20÷4, we can ask, “How many 4’s are in 20?” The process is to keep subtracting 4 (using concrete, pictorial, and abstract methods) until zero is reached. This would be done 5 times — thus, 20 ÷ 4 = 5.

Much like multiplication, there are different aspects of division children should become familiar with.

- Arrays
- Equal Groups
**Repeated Subtraction****Number lines**- Skip counting

The focus today will be to help children understand how repeated subtraction can assist with the division process (using manipulatives, drawings, and paper-pencil methods). The template pictured here is FREE from: Multip. and Division templates FREE from Number Two Pencils @ TpT

The reason the repeated subtraction strategy is important is because this is what we are really asking students to do when they encounter long division or partial quotient problems. With the problem 100 ÷ 4, the question is, “How many 4’s are in 100?” If the repeated subtraction process is used, the answer is of course, 25. But subtracting 4 twenty-five times is not very efficient. So we want the student to get closer to 100 and subtract larger amounts than 4 at a time. The partial quotients method would allow the student to do this in chunks. 1 solution could be to subtract 40 (ten 4’s), subtract another 40 (ten more 4’s), subtract 20 (five 4’s). See picture below:

With the std. long division algorithm, students also must think of division as repeated subtraction. You wouldn’t believe the number of “ah-ha’s” I get from students when I show them this concept!! Here’s a link to a previous post on long division. Making sense of division

So, how do we encourage this strategy with new learners using basic division facts? Check these out:

**Manipulatives:**If the problem is 20 ÷ 4, start with 20 objects. Then take away 4 at a time by moving them to the side and forming a group. Repeat until all 20 objects have been taken from the original group. How many groups were made? This is a quick, meaningful any very efficient method to help students actually see equal groups being constructed. A**must**before going to drawings or paper pencil methods.**Drawings:**If the problem is 15 ÷ 3, draw 20 objects. Cross off 3 at a time and keep track (ex: tallies). Keep crossing off 3 at a time until all 15 objects have been crossed off. How many times was this done?**Number line:**If the problem is 10 ÷ 2, draw an open # line and label with 10 points (0-10). Starting with 10, jump backward 2 at a time until 0 is reached. Count how many jumps were made. This is more effective with dividends 20 or less due to space and amount of time this takes. Consider a vertical # line as well.**Subtraction:**If the problem is 56 ÷ 8, write the # 56, then subtract 8 repeatedly until there is zero left. Keep a running total. Common errors with this method are failure to follow regrouping steps and poor calculations.

I came across this FREE resource from Number Two Pencils: Free Multip. and Division templates showing a template for illustrating the 5 division strategies. I completed two of them for you to see how these can be used effectively in the classroom. They could also be used as anchor charts (and then student can create some of their own). There is also a companion set for multiplication strategies. Get your set here: Multip. and Div. strategy anchor charts

**Enjoy your division journey! Stay tuned for more!**

Last post featured division using arrays and the area model. This post will focus on helping children see division as **equal groups**. Most of us have used the “plates of cookies” analogy to help kids see how to represent equal groups in a drawing. I will just take that a few more steps to increase efficiency.

Much like multiplication, there are different aspects of division children should get familiar with:

- Arrays
**Equal Groups**- Repeated Subtraction
- Number lines
- Skip counting

In this post, I will break down the benefits of equal groups models to help children understand division (and how it is related to multiplication). Check out the freebies within this post.

If you haven’t utilized this book with your students, please try to find a copy! It’s called The Doorbell Rang by Pat Hutchins. In this story, Ma makes some cookies to be split between the kids. Then the doorbell rings and more kids come, so the problem has to be refigured. This scenario repeats. As a class, you can duplicate the story with a different # of cookies and children.

Another great story emphasizing equal groups (as well as arrays) is the story One Hundred Hungry Ants by Elinor Pinczes. In this story, 100 ants are on their way to raid a picnic. They start off in one straight line (1 x 100), but then rearrange into different equal groups to shorten the line (2 lines of 50, 4 lines of 25, etc.). A nice project after reading this book is to see how many ways a different given # of ants (or other animals / objects) can be divided into equal groups / rows.

By clicking on the links for each book above, you will be taken to Amazon for more details.

As I mentioned earlier, many children’s view of equal groups regarding division is to use manipulatives and/or draw circles / plates to match the divisor and then divide up the “cookies” equally in these groups. Let’s say you had this problem: “There are 12 cookies to be divided onto 3 plates equally. How many cookies would go on each plate?” As you observe the students:

- How are they dividing up the cookies? One at a time, two at a time, randomly, trial and error?
- Are the “cookies” scattered randomly on the plate / circle? Or, are they arranged in an easy-to-see pattern so they are easily counted (by the student and yourself as you walk around the room)?
- Are the students able to verbally tell you
**how**they divided them? - Are the students making the connection to multiplication by noting that 3 x 4 = 12?
- Can they solve similar problems using language other than plates / cookies?
- Try shelves / books; trays / brownies; buildings / windows; flowers / petals; students / rows of desks, stars / # of points; aquariums / fish; boxes / donuts; etc.

**Use of manipulatives of various types (cubes, tiles, counters) is important for children to have their hands on the objects being divided. This is how they work out their thinking. Then work toward paper/pencil drawings before going to the abstract use of numbers only. **

Also, help children list **synonyms** for the dividing process: *distribute, share, split, separate, halve, quarter, partition*

**Here are a few strategies I believe help make the equal groups process more efficient:**

- When using manipulatives or drawings, instead of
*randomly*placing the objects being divided into equal groups, arrange them so it’s easy for the child as well as the teacher to see at a glance how many there are. In other words, if there are 5 in each group, are they randomly scattered? If they are, the child wastes a lot of time recounting, which often invites error. And the teacher has to spend time rechecking the child as well. Or, are the objects arranged in smaller arrays or groups making it very easy to see the total (like dice? by twos?). This little requirement adds to a child’s understanding of number bonds and multiplication. - Instead of placing individual objects, have the students try tally marks. Again, these are counted more efficiently than a random organization – and it aids in multiplication.
- Instead of always using a one-at-a-time strategy as objects are being distributed, help them think that often they can try 2 at at time, or 5 at a time. This aids with estimation and helps transfer this idea to future long division processes – especially partial quotients.
- Connect use of manipulatives and drawings with real life stories. What things come in equal groups?
Refer to one of my previous blog posts showing this template for stories about equal groups (which can be multiplication or division): Equal Groups blog post. Help students notice each problem consists of these three components.

- # of groups
- # in each group, and
- the total #
- The division problem will usually provide the total and one of these (# of groups; # in each group). So the problem will be to determine the missing component by relating known multiplication facts and/or dividing.

**Stay tuned! Next week I will include some helpful basic division concepts resources.**

Division seems to be the hot topic with classes I have been visiting lately, so I thought I’d focus on that for now. Let’s look at some of the basics. Students as young as first grade actually start thinking about division when working on fraction standards such as: Determine fair share — equal parts. Most students have had practical experience with dividing sets of objects in their real life to share with friends, classmates, or family (cookies, pizza, crayons, money, pieces of paper). So now our job as teachers is to relate this real-life experience with the division algorithm.

Much like multiplication, there are different aspects of division children should get familiar with:

**Arrays**- Equal Groups
- Repeated Subtraction
- Number lines
- Skip counting

In this post, I will break down the benefits and uses for arrays (and the related area model) to help children understand division (and how it is related to multiplication). There’s a fun **FREE** game (Block-It) at the end of the post.

**What is an array?** An array is a rectangular model made up of **rows** and **columns**. When an array is constructed, the **factors** are represented by the number of rows and columns. So, do your students know the difference in a row and column? (Rows go **horizontally**, while columns are **vertical**.) These are important math terms students should be using.

- Give students experience
*constructing*arrays with manipulative objects (tiles, chips, cubes, etc.):- You can be specific, such as: “Build an array using a total of 12 tiles. Put them in 3 rows. How many columns did you create?” In this scenario, there is only 1 way to show this array. Students would be modeling 12 ÷ 3 = 4. Twelve is the
**dividend**(the total amount you started with). The # of rows is the**divisor**(*how*it was divided). The**quotient**is the result (in this case the # of columns). - You can also be a little more open ended such as: “Build an array using 12 tiles. Is there more than one way to do this?” If students are given the opportunity to explore, they hopefully find arrays such as 3 x 4; 4 x 3; 2 x 6; 6 x 2; 1 x 12; or 12 x 1. Students would be modeling 12 ÷ 4; 12 ÷ 2; 12 ÷1, etc.

- You can be specific, such as: “Build an array using a total of 12 tiles. Put them in 3 rows. How many columns did you create?” In this scenario, there is only 1 way to show this array. Students would be modeling 12 ÷ 3 = 4. Twelve is the
- Give students experience
*drawing*arrays:- You can be specific or open-ended (as above).
- Children can free-hand draw or use grid paper. If using grid paper, then these can be cut out and displayed as “Different ways to divide 12.”

- Give students experience using
*pre-drawn arrays*:- Students should label the sides of the array with numbers.
- Use the numbers shown to determine the fact family. Example: 3 x 4 = 12; 4 x 3 = 12; 12 ÷ 3 = 4; and 12 ÷ 4 = 3

- After the array is made, ask questions or explore more such as:
- How many 3’s are in 12? (count the columns)
- How many 4’s are in 12? (count the rows)
- Circle the rows and / or columns to see the groups more easily.
- Help children make up story problems to match the array: “I have 12 desks that I need to arrange in 3 rows. How many desks will be in each row?” or “I need to put 12 books equally onto 3 shelves. How many books will go on each shelf?

Relate experience with arrays to determine **area** of a rectangle. This mostly just adds a measurement component to the problem.

- Draw a rectangle and partition it into columns (length) and rows (width) to match the story. Here are two sample stories:
- I am making a rectangular shaped garden which I want to be 12 square yards in size. If the length of the garden is 4 yards, how long does the side of the garden need to be?
- I am using a rectangular piece of wood to cover a broken window that is 12 square feet. One side of the wood is 3 feet. How long would the adjoining side be?

**Here’s a great game called “Block it” which utilizes arrays. It can have variations depending on the level of your students. Here is a FREE copy of the directions:** Block-It Game Directions

Materials needed:

- 1 grid paper (1/2″ is great)
- 2 players
- Each player needs 1 crayon or colored pencil (light colored). Different color per player.
- 2 number cubes or dice (6 sided).

How to play:

- Player 1 rolls the dice. Let’s say a 3 and 4 are rolled. The player makes a 3 x 4 “block” or array. Be sure to show them how to use the lines on the grid paper to make this (as I discovered it’s not always clear to some students). Color it in with crayon. Inside the block, write the product (12).
- Player 2 then rolls the dice and uses their 2 numbers to create another block, colors it, labels it, etc.
- Repeat
- The goal is to create as many blocks / arrays as possible (more than the opponent). There is a strategy to maximize the use of the space. Repeated play helps children see they need to consider this so they don’t end up with little unusable spaces.
- As the board gets filled up, players may have to miss a turn or roll again if not enough space is available.

Variations:

- As the board gets filled up, students may need to start thinking of alternate ways to make their blocks to fit the available space. For example, if the player rolls a 6 and 4 but there is no room to fit a 6 by 4 array, they can think of other ways to make an array of 24 that might work (such as 8 x 3, 12 x 2).
- Students can keep track of their score by keeping a running total of each block / array they make.
- Use smaller size grid paper and use 9, 10, or 12 sided dice.
- Write the fact family members for each block created.

**Enjoy! Have you / your students played Block It? Let us know if you like it! **

Today’s post is the result of a project I have been working on for awhile. I created some posters you can use in your classroom which feature sentence frames connecting text structure to the skills of main idea and summarizing.

Here are samples of 2 of the Main Idea posters. Get the full set here FREE: Text Structure Main Idea Posters CE-2019 There are 10 posters (1 Main Idea and 1 Summarizing poster for each of the five text structures). If you have suggestions for improvement, please let me know. I want to make these usable for YOU!

]]>Welcome back to the third text structure post. Today’s focus will be on sequence / chronological order and descriptive text structures. Here are some graphic organizers to keep in mind.

**Sequence / Chronological Order**

1. Sequence refers to a particular order in time. This can be:

- Information presented minute by minute, hourly, weekly, monthly, yearly, etc.
- Providing information by dates (a timeline)
- Steps of how to complete something (first, second, third, etc.)
- A retelling of events in the order they happened: First, next, then, finally or beginning / middle / end. It may be helpful to use a “retelling rope”. Use a section of rope or nylon cord (approx. 1 foot long). Tie several knots along the length of it (3-5). At each knot, retell part of the story or events in sequence.
- Observing how things / people have changed over time
- Non-fiction and fiction selections
- Arranging events in order using pictures

2. Connecting sequence to strategies:

*Predict*what will happen next in the sequence.*Visualize*the steps involved.*Make personal connections*regarding your own experience with the sequenced topic.

3. Sequence / Chronological order main idea / summarizing sentence frames: Suppose I read an article telling about the seasonal journey of a pod of whales. Again, the topic is *whales* — but this is NOT the main idea.

- (Main idea): Whales travel to different locations each season to find food and a mate.
- How to ________ step by step.
- The timeline of _________________.
- There are several steps to ______________. First, _________. Then, ___________. Last, ________.
- The life cycle of __________.
- Many things happened during _____________’s life.
- (Summarize): Whales travel to different locations each season to find food and a mate. In the spring, they ________. In the summer, ______________. In the fall, _____________. In the winter, _________.
- To make ________, follow these steps: ________________.
- The life cycle of a ___________ includes these stages: _______________.
- Many things happened during _____________’s life. In (year), he/she_____________. After that, _____________. Then, ________________. Finally, ___________________.

**Descriptive Text Structure**

1. Descriptive structures give details. These can be:

- Details or descriptions about a person, a place, a thing, an idea, an animal, an event, etc.
- A web graphic organizer is a good model to visualize, with the topic in the center and the supporting details branching outwards.

2. Connecting to strategies:

*Visualize*what is being described, especially if there are no pictures or photos in the text.*Ask questions*about the topic such as: “I wonder . . .”*Analyze the point of view:*What is the author’s point of view. Is he/she presenting a one-sided view of the details presented?*Make connections*to the topic.

3. Description text structure main idea / summarizing sentence frames: Suppose I read an article telling interesting facts about whales (species, its habitat, what it eats, size, what it preys on, etc.). Yes, the topic is *whales*, but this is NOT the main idea.

- (Main idea): There are many interesting facts about whales.
- There are many interesting facts about _________________.
- This article tells about the ___________ of ________________.
- (Summarize): There are many interesting facts about whales, such as _________________.
- This article tells about the __________ of ______________. Some interesting facts include __________________ and _______________.

Pictured below is an anchor chart I made showing some easy graphic organizers and some common key words regarding all of the text structures. Focus more on the structure than the key words. Utilize the graphic organizer to help students take notes, and then use the information from the graphic organizer to state the main idea and summarize.

**Resources for Sequence or Descriptive text structures:**

- Reading Rockets: sequence with list of books
- Librarything.com: List of sequencing stories
- Sequence graphic organizer (free at TPT)
- Text Structure graphic organizers (Free on TPT) * This is the source of the graphics at the beginning of this post.
- Text Structure Tic-Tac-Toe ($1.99 from Ms To-do-list at TPT)

Enjoy! Looking forward to your comments about text structure! In part 4, I will share some text structure posters with you.

]]>Welcome back to part 2 regarding Text Structure. As I mentioned before, pairing a text with a graphic organizer to help highlight the structure can be very helpful to frame the main idea and summary. When a graphic organizer is used often, then students begin to visualize them and organize their thoughts mentally as well. And still better . . . combining text structure instruction with reading strategies such as visualizing, questioning, making connections, and predicting will lead to higher comprehension.

Today’s focus will be on two other text structures: Cause / Effect and Problem / Solution. These two are related, but often confusing to students. Look for some resources at the end of this post.

**Cause and Effect:**

Cause: The reason why something happened.

Effect: The result — what happened?

A cause / effect text structure can show 1 cause and several effects. Example: An earthquake can be the cause of many events (damaged structures, ruptured pipes, injuries, accidents, tsunami, etc.). When this is the case, it may be simpler to identify the cause first, then identify all of the effects.

On the other hand, a cause / effect text structure can show several causes for 1 effect. Example: Some animals are endangered (effect) due to these causes: pollution, loss of home environment due to destruction of their habitat, weather, disease. When this is the case, it may be simpler to identify the effect first, then identify all of the causes.

**Other notes about teaching cause / effect:**

- This text structure can apply to non-fiction as well as fiction texts.
- Because many cause / effect relationships require defining the problem (which could be the cause and sometimes the effect as well), students often get confused and identify the structure as problem / solution.
- Not all cause / effect relationships are about problems. Example: I love my grandson’s drawings (cause), so I hang them on the refrigerator (the result / effect). No problem here!
- While most anchor charts posted online provide key words for the cause / effect structure (because, reason, since, as a result, etc.), I would suggest limited use of them especially when first analyzing the structure. I have found when mentioning them first, students often just start looking for those key words and are not truly reading the text. And . . . those words can also be found in almost any text anyway. You don’t want kids to reduce this to a competition: “How many time did I find the word
*because*?” Those words don’t even*have*to be there for there to be a cause / effect relationship. - Use a graphic organizer with an arrow connecting the cause to the effect.
- Even young students can understand simple cause / effect relationships presented in stories. Discuss the causes and effects and/or write them as a shared writing experience. See some resources below for great books on this structure.

**Combining with strategy work:**

*Visualize*actions of the subjects in the text to*picture*the causes and results.*Make connections*to things, places, events in the text you have experienced. Make*predictions*based on those experiences regarding why things happened.- Help students
*ask questions*about the text. They should be wondering why certain things happen, or what caused what. Learn to read on (or check other resources) to see if those questions get answered. - Make
*inferences*about the causes in the text. Read between the lines.

**Connecting to main idea and summary.** Supply some sentence frames so students are using compare/contrast language. Suppose an article describes the **causes** of beached whales. The topic is whales — but that’s NOT the main idea:

- (Main Idea): There are many
**reasons**a whale becomes beached. - (Summary): There are many reasons a whale becomes beached such as low tide, changes in ocean currents, chemicals in ocean water, and disorientation due to man-made sonar devices.
- (Main Idea): There are many
**causes**for _________________________. - (Main Idea): The main cause for ____________ is _______________.
- (Main Idea): There are several
**reasons**why __________ decided to ___________. - (Summary): There are many causes for __________________ such as _________________.
- (Summary): When _______________ happens, the
**result**(s) are ___________________.

**Problem / Solution Text Structure:**

This structure is very similar to cause / effect. A problem is identified. Most likely there is also a cause to the problem. But in this case, a solution is proposed or acted on.

**Notes about teaching problem / solution text structure:**

- This text structure is also common in fiction stories. Conflict and resolution are common in scenarios such as character vs. character, character vs. self, character vs. society, character vs. nature, etc.
- Emphasis may be put more on
*how*the problem should be or was solved. - Often times multiple solutions to a problem are offered.
- Use a graphic organizer such as completed puzzle pieces to symbolize the solution aspect, or a light bulb idea bubble.
- Compare the author’s point of view regarding the solution to your students points of view. Discuss the pros and cons of the proposed solutions.
- See resources below for some good texts regarding problem / solution.

**Connecting to main idea / summary.** Suppose an article tells about **problem** encountered whale watching and **how to** avoid them. The topic = whales —- but that’s NOT the main idea.

- (Main Idea): To avoid problems encountered during whale watching, you should learn about their paths and habits.
- (Summary): To avoid problems encountered during whale watching, the observers can do things such as watch from a distance, stay parallel to them, and respect their space.
- (Main Idea): A
**solution**to the**problem**of ___________ is ______________. - (Summary): In this article, the
**problem**was ____________. It was**solved**by ___________.

**Resources I recommend:**

- Consult readworks.org. There is a good article (at 4th grade level) regarding cause /effect about the reasons frogs in South America are dying. You will need to join, but it is free (and a GREAT resource).
- Also at readworks.org, a good article (at 3rd grade level) regarding problem / solution. It describes how one state had too many sheep and one state didn’t have enough moose, so a solution to make a switch was organized.
- Click here for some books with cause / effect structure
- Click here for some books with a problem / solution structure
- Cause / effect activity (FREE from Mrs. Patmore @ TPT):Cause and Effect Detectives
- Cause / effect task cards (FREE from Foreman Teaches @ TPT): ID the cause and the effect task cards

**Stay tuned for more text structure information. Please feel free to comment and share your ideas as well.**

I have come to realize just how important knowledge of text structures is to almost all of the other comprehension skills and strategies. So that will be my focus for the next few posts — how this text structure connection relates to **main idea, summarizing, note-taking, and writing**. This post will feature the compare and contrast text structure (and some resources at the end of this post).

**What are the text structures?** Most sources consider the following 5: (Picture from Mrs. M’s Style. Here’s the link on Pinterest: Text Structure Mini Anchor Chart)

- Compare and Contrast
- Cause / Effect
- Sequence
- Details / Description
- Problem / Solution

When I see reading texts that indicate the *week’s* skill is text structure, I cringe a little bit. Why? Well, if you are teaching all 5 of them – that’s too much to digest in one week. Here’s what I think is much more practical: Teaching about text structures should occur with each and every reading selection — and refer to the structure that is most evident regarding that selection.

Here’s an example of what the teacher might say: “This week we are reading an article titled Whales and Dolphins. This article will **compare and contrast** whales with dolphins. **Compare and contras**t is a text structure in which the author will tell ways the whales and dolphins are alike and different from each other.”

**How can I further connect this to comprehension and text structure?**

- Venn Diagrams or T-charts are helpful
**graphic organizers**regarding compare/contrast text structure. Student can take notes using the graphic organizer. The idea is that with frequent use, students can eventually visualize this graphic organizer model in their head. Then this visual model serves as a thought organizer when they are not able to physically utilize one. - I can direct my questions to
**focus on this text structure**such as: “On page 37, can you find one way the author**compared**whales to dolphins?” “On page 39, the author told 3 ways the whales and dolphins are**different**. What did he say?”

**How can I further connect this to help students with the main idea and/or a summary of a compare/contrast article? ** Using information from notes on the Venn Diagram, students can use sentence frames like these:

- This article compared _____________ to ______________. (main idea)
- This article compared ___________ to _____________. Whales and dolphins are alike because _____________ and they are different because ___________________. (summary)

**How can I further connect this text structure to writing?**

- Students use their graphic organizer notes to write complete sentences. “While whales and dolphins both live in the ocean, there are many other ways they are alike and different. . . .”

Using a graphic organizer (as mentioned above) is also a good strategy to encourage **student engagement**. Writing down the information gives two ways to recall it (mentally and in writing). If the student works with a partner to do this, they have a third way (via speaking and discussing). Then coming together with a larger group to compare notes increases student engagement and lets them hear other students’ viewpoints.

Finally, try to **connect this skill to other strategies** (see my previous post on *Comprehension Strategies).*

- Use the
*visualize*strategy to compare and contrast characters, events, or items from a text selection. - Use the
*making connections*strategy to compare and contrast points of view, or yourself with a character in the story. - Use the
*predicting*strategy to compare and contrast how different events or characters will change throughout the text.

**Resources:**

- Here is one of my favorite text structure articles from Reading Rockets. Implementing Text Structure Strategies
- This is one of my favorite text structure graphic organizers (FREE from Kristi Dunckelman – Pelicans and Pipsqueaks at TPT): Non-Fiction Text Structures. The graphics above are from her source.
- A great story for compare and contrast: Alexander and the Wind-up Mouse by Leo Lionni (compare a real mouse with a wind-up mouse)
- A great article featuring information on comparing different versions of fairy tales, books to movies, books by same author, etc (by Read Brightly): How to use books to teach comparative thinking

In the next few posts, I will share some other resources I feel are great for Text Structure. STAY TUNED!

]]>This post features 3 more area and perimeter misconceptions students often have. I have included some strategies using concrete and pictorial models to reinforce the geometry and measurement standards. Refer to Geometry Part 7 for 2 other common misconceptions.

**Also, check out some free resources at the end of this post!!**

**Misconception #3:**** **A student only sees 2 given numbers on a picture of a rectangle and doesn’t know whether to add them or multiply them.

- Problem: The student doesn’t know the properties of a rectangle that apply to this situation — that opposite sides are equal in measurement.
- Problem: The student doesn’t see how counting squares can help calculate the area as well as the perimeter.

Ideas:

- Give the correct definition of a rectangle: A quadrilateral (4 sides) with 4 right angles and opposite sides are equal.
- Give the correct definition of a square: A quadrilateral (4 sides) with 4 right angles and all sides are equal. From this, students should note that
*squares are considered a special kind of rectangle.*Yes, opposite sides are equal – but in this case all sides are equal. - Using square tiles and graph paper (concrete experience), prove that opposite sides of a rectangle and square are equal.
- Move to the pictorial stage by making drawings of rectangles and squares. Give 2 dimensions (length and width) and have students tell the other 2 dimensions. Ask, “How do you know?” You want them to be able to repeat “Opposite sides of a rectangle are equal.” With this information, students can now figure the area as well as the perimeter.
- Move to the abstract stage by using story problems such as this: Mr. Smith is making a garden. It will be 12 feet in length and have a width of 8 feet. How much fence would he need to put around it? (
*perimeter*) How much land will be used for the garden? (*area*). - Measure rectangular objects in the classroom with some square units. Show how to use them to find the perimeter as well as the area using just 2 dimensions. Ask, “Do I need to fill it all the way in to determine the answer?” At the beginning – YES (so students can visualize the point you are trying to make). Later, they will learn WHY they only need to know 2 of the dimensions to figure the area or perimeter.

**Misconception #4:**** **A student hears this: “Record your measurement for area as *square inches* and the measurement for perimeter as *inches.” Note: This applies to use of units such as cm, feet, meters, yards, miles, and so on.*

- Problem: The student doesn’t understand the difference between
*square*and non-square measurements.

Ideas:

- Show objects with a measure of one square unit. Some possibilities:
- One cm or square inch graph paper
- Cut some construction paper to measure 1 foot by 1 foot so students see what a square foot looks like. Put 9 together to form a square yard. Use it with some things in your classroom (door, bulletin board, desk top, table, etc.)
- Look at a map of your town to see 1 square mile in many square blocked neighborhoods.

- Use a ruler, tape measure, or string as another way to measure the perimeter of objects. In this way, they can see these items are not made of squares.
- Show shapes such as these (no squares visible). If you ask students to put them in order from least to greatest by their size, how could this be accomplished? (Remember, some students don’t have conservation abilities and think longer or taller is always more.) The only real way to determine size is to measure the area of the inside.
- Use the often forgotten geoboard and geobands (rubber bands) to create different areas (the squares) and perimeters (the rubber bands). Here is the link to an app for a virtual geoboard: Math Learning Center geoboard app (Note: I couldn’t get the rubber bands to work, so I used the pencil tool to draw the outlines.)

**Misconception #5:**** **Students think there may be a relationship between area and perimeter. They may think all shapes with the same area have the same perimeter.

- Problem: This means if one shape has an area of 12 square inches, and the perimeter is 16 inches, they might think all shapes with an area of 12 have perimeters of 16 inches.

Ideas:

- The geoboard activity above is a way for students to see that shapes with the same area can have different perimeters.
- See picture below: Using tiles and/or graph paper, assign each group or pair of students a number with the directions to create a shape with that perimeter. Caution: connecting squares must match edge-to-edge (not tip-to-tip) with no holes inside the shape. Once they make different shapes with the same perimeter, they can also determine the area. They will notice shapes with the same perimeter don’t necessarily have the same area. Students enjoy the challenge to see all of the different ways they can make a shape with the given perimeter.
Here are some area and perimeter resources I like:

- The geoboard app mentioned above (free): Math Learning Center virtual geoboard
- K-5mathteachingresources.com (free) Fencing a Garden problem
- K-5mathteachingresources.com (free) Designing a Zoo Enclosure problem
- K-5mathteachingresources.com (free) Square Units
- K-5mathteachingresources.com (free) Jack’s Rectangles problem (area and distrib. property)
- K-5mathteachingresources.com (free) Area of Rectilinear Figures
- K-5mathteachingresources.com (free) Rectangular Robot (perimeter activity)
- Laura Candler (free @ TPT) Mystery Perimeters (great for determining missing dimensions on irregular rectangular shapes)
- Math Coachs Corner ($5 at TPT) Area and perimeter task cards

Today’s topic is the measurement of area and perimeter. Even though these may be considered measurement standards, they are highly connected to geometry (such as the attributes of a rectangle). Check out previous posts in my Geometry series (Composing and Decomposing) for other mentions of area and perimeter.

Misconceptions provide a window into a child’s thinking. If we know the misconceptions ahead of time, we can steer our teaching and directions to help students avoid them. I will go through several misconceptions and some strategies and/or lessons that might address them. Misconceptions #1-2 appear in this post. Misconceptions #3-5 will be featured in next week’s post.

**Misconception #1:**** **A student hears this: “We use area to measure inside a shape and perimeter to measure around a shape.”

- Problem: The student doesn’t know how to apply this definition to real situations which require the measurement of area and/or perimeter.
- Problem: The student may think, “Since perimeter measures the outside edge, then area means to measure the inside edge.”
- Problem: Students confuse the two terms.

Ideas:

- Brainstorm with students (with your guidance) examples of the need for area and perimeter. Use these scenarios as you solve concrete or pictorial examples.
- Area: garden, room size, ceiling tiles, carpet, rug, floor tiles, football field, tv screen, wallpaper, wall paint, etc.
- Perimeter: picture frame, fencing, floor trim, wallpaper border, bulletin board border . . .

- Show this diagram which emphasizes the concept that area measures the entire inside surface using squares of different sizes (cm, inch, foot, yard, mile), while pe
**rim**eter measures the rim / edge / around an object or shape usually using a ruler, tape measure, or string. - Try this project: Use graph paper and one inch tiles (color tiles) to
**concretely**make shapes with a given area. Example, “Build a rectangle with an area of 24 tiles.” Specify there can be no holes in the rectangle — it must be solid. Students can experiment while moving the tiles around. Then trace the rectangle and cut it out. With this practice, students are also focusing on arrays and multiplication. Did they find 4 ways? (2 x 12, 3 x 8, 4 x 6, 1 x 24)? Did they find out a 3 x 8 rectangle is the same as an 8 x 3 rectangle (commutative property)?- Note: Make sure students stay on the given lines when tracing their shapes and cutting. I found this to be difficult for many students even though I said explicitly to “Stay on the lines when you trace.”

- Similar to the above: Use the tiles and graph paper to create irregular shapes with a given area – meaning they don’t have to be rectangular. Give the directive that tiles must match at least one edge with another edge (no tip to tip accepted). And, same as above — no holes in the shape. You can even assign different areas to each small group. Compare shapes – put on a poster or bulletin board.
- Using the same shapes made above, determine the perimeter. I suggest placing tick mark to keep track of what was counted. With this lesson, students will hopefully realize shapes with the same area do not necessarily have the same perimeter. Advise caution when counting corners or insets – students usually miscount these places.
- Try this project: Design a bedroom with dimensions of (or square feet of): ___________ Include a bed, and at least one other piece of furniture or feature (nightstand, dresser, desk, shelf, chair, closet, etc.)
- The student can use smaller scale graph paper with 1 square representing 1 square foot.
- Let them have time to “work it out” and practice perseverance as a rough draft before making their final copy.
- Together, use the square foot construction paper pieces to determine an appropriate size for a bed and other furniture. Students must think of it from an overhead perspective. When I did this with a class, we settled on 3 ft. x 6 ft bed for a total of 18 square feet. We did this by laying the square foot papers on the floor to see in real life what this size looked like.
- Label the Area and Perimeter of each item in the bedroom.
- The items in the bedroom could also be made as separate cut-outs and arranged / rearranged on the “floor plan” to see all of the different ways the room could look.
- Students in 3rd grade might want to use whole units, while 4th and up might be able to use half-units.

- On a test, area answer choices would include ” ____
**square inches**” or “**inches squared”**or “**inches²”**Answer choices for perimeter will omit the word “square.”

**Misconception #2**: A student hears this: “To find area, multiply the length times the width.”

- Problem: Student doesn’t know their multiplication facts.
- Problem: Student doesn’t know which dimensions are the length and width.

Ideas:

- A good guide: The
**l**ength is the**l**ongest side. The width is the shortest side. - The length and width are two
**adjoining**sides (not opposite sides). - Show how to partition a rectangle into squares. If a rectangle had a dimension of 4″ x 2″, then show them how to make 4 columns and 2 rows. Watch how they do this. For many students, they would draw 4 vertical lines inside to make the columns and 2 horizontal lines to make the rows. This would obviously result in a 5 x 3 set of squares – but students don’t always check. Do they know — “To divide a rectangle into 4 columns, I only need to draw 3 lines.” Now, counting squares isn’t the most efficient method – but one that might help students who struggle with the concept or with multiplication facts.
- Some rectangles are too large to draw squares in. If students don’t know their multiplication facts, the rectangle can be partitioned into 2 (or more) smaller rectangles using facts they do know. Then the area of each smaller rectangle is added together to find the total area. See
**picture below**for an example. Again, students have to connect to geometry to understand that opposite sides of a rectangle are equal.- Example: A rectangle has dimensions of 8 x 7. Using facts more readily known, break one of the dimensions into 2 addends (such as breaking 7 apart into 5 + 2). Partition the rectangle into 2 rectangles and use the 5 + 2 to label one side (instead of 7). Using the concept of the distributive property, the student is calculating this: 8 x 7 = 8 (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56 sq. units.

**Misconception #3:**** **A student only sees 2 given numbers on a picture of a rectangle and doesn’t know whether to add them or multiply them.

- Problem: The student doesn’t know the properties of a rectangle that apply to this situation — that opposite sides are equal in measurement.
- Problem: The student doesn’t see how counting squares can help calculate the area as well as the perimeter.

**Misconception #4:**** **A student hears this: “Record your measurement for area as *square inches* and the measurement for perimeter as *inches.” Note: This applies to use of units such as cm, feet, meters, yards, miles, and so on.*

- Problem: The student doesn’t understand the difference between
*square*and non-square measurements.

**Misconception #5:**** **Students think there may be a relationship between area and perimeter. They may think all shapes with the same area have the same perimeter.

- Problem: This means if one shape has an area of 12 square inches, and the perimeter is 16 inches, they might think all shapes with an area of 12 have perimeters of 16 inches.

**Next post: Ideas to address Misconceptions #3, 4, and 5 and links for area and perimeter activities!! STAY TUNED**

When working with students on geometry lessons involving angles and lines, I notice many **misconceptions. **So . . . I thought I would share them with you on this post. Some activity ideas and freebies are located at the end of this post.

**Right Angles: **

- Students can only see the 90° angle if it is presented in the direction as a capital L.
- Try turning the angles in different positions.
- It is still considered a “right” angle even though it is turned to the left, up, or down.

- Students are told if they can draw a square inside the angle, then it is a right angle. So if it looks “squarish” to them, they think it’s a right angle.
- Show them how to put the square corner of a piece of paper or index card into the angle to check. Take time to have them practice – don’t assume they know how.

**Acute Angles (angles less than 90°):**

- Students often can’t tell if the angle is <90° if it is oriented upside down or if one of the rays is not aligned horizontally.
- Show how to put the square corner of a piece of paper into the angle to check. If the paper covers up the angle, it is <90°.

- Students are told an acute angle “is a
*cute*little angle.” I am guilty of having done this in the past. But if a student sees*any*angle made up of short lines, they may interpret it as “little” or acute.- Remind them it’s the size of the angle that makes it acute, not the size of the lines.

- While right angles are exactly 90°, students may expect an acute angle to be given a specific number.
- Acute angles range from 1° to 89°.

**Obtuse Angles (angles greater than 90°):**

- Students often can’t tell if the angle is >90° if it is oriented upside down or if one of the rays is not aligned horizontally.
- Show how to put the square corner of a piece of paper into the angle to check. If the ray is wider than the paper corner, it is >90°.

- Students are told an obtuse angle is big, and may classify an angle with long rays as obtuse (because it is a large picture).
- Remind them it’s the size of the angle that makes it obtuse, not the size of the lines.

- While right angles are exactly 90°, students may expect an obtuse angle to be given a specific number.
- Obtuse angles range from 91° to 179°.

**Lines: Parallel, Perpendicular, Intersecting**

- Students get the terms
*parallel*and*perpendicular*confused.- Notice the word “parallel” has a set of parallel lines in the word (the two l’s).

- Students may see two lines and assume they are parallel just because they are
*not*intersecting.- Remind them that parallel lines are those who are the same distance apart.
- Show that 2 non-intersecting, non-parallel lines would intersect if you extended their lines.

- Students think lines must be intersecting OR perpendicular.
- Show them that all perpendicular lines are
*also*intersecting lines, but not all intersecting lines are perpendicular. - Perpendicular lines are a special type of intersecting line (with 90° angles).

- Show them that all perpendicular lines are

**Some good hands-on activities:**

- Locate angles and lines of all types in the classroom.
- Bring in pictures and/or photographs of different lines and angles for collages or posters. Architectural features would be an excellent source.
- Create them with toothpicks, chenille stems, craft sticks, paper strips with brads, etc. BUT, help students see that acute and obtuse angles can be turned different directions and come in different sizes (acute angles range from 1° to 89° and obtuse range from 91° to 179°).
- Using an analog clock, name times in which the hour and minute hands form acute angles, right angles, obtuse angles, and straight angles. Here are some examples:
- Acute angles: 1:00, 2:00, 6:20
- Obtuse angles: 4:00, 8:00, 1:30
- Right angles: 3:00, 9:00, 12:15 (approximately)
- Straight angles: 6:00, 12:30 (approximately — or 12:32 exactly due to hour hand placement)

- Use a map of the United States to classify states according to the type of angles in their borders.
- Which states have right angles?
- Which states have acute angles?
- Which states have obtuse angles?
- Do any states have all 3 types?

- Using the capital letters of the printed standard alphabet, classify them by the types of lines
*and*angles. Provide individual copies of the printed alphabet to each student (or pair of students). Click here for a**FREE**recording sheet: Classify Letters of the Alphabet CE: - Here is a
**FREE**measuring angles partner game called “Tangling with Angles .” A student draws an angle. Partner names the type of angle and estimates the measurement. Points are earned based on the difference between the estimate and the actual measurement. The partner with the least number of points is the winner. - This is a good set of posters by Kathleen and Mande Lines and Angles Posters $2.25 @ TPT

**Share some of your favorite lines and angles teaching points or activities!!**