I am teaching dividing fractions now and wanted to be clear in my own mind about the differences between these two ways of thinking about and describing division problems. I made this graphic to help me. I find quotative division very helpful when describing fraction division problems. For example, how many 1/2s are in 10 wholes? However, partitive division makes more sense when the dividend is a fraction or mixed number and the divisor is a whole number. For example, 4 3/5 lbs of candy is divided among 6 friends. How much candy does each friend get?

# Math to the 7th Power

## Wednesday, November 15, 2017

## Saturday, September 16, 2017

### Factors and Multiples Puzzle from NRICH

This Factors and Multiples Puzzle from NRICH was a perfect way to start the school year with my sixth grade students. It is one of the best low threshold high ceiling tasks I've used and creates opportunities for students to work together and discuss numbers and sets of numbers while they review essential vocabulary in the context of an interesting challenge.

Here are some things that I did the day before we worked on the puzzle that I think helped students get started right away and work on the puzzle for the entire class period.

1) I did notice and wonder with square numbers and triangular numbers using these graphics which I recreated based on inspiration from this poster

2) I did some mini-challenges where students, using small whiteboards, wrote down as many numbers as they could think of that meet two conditions. Students loved this!

- Think of numbers that are
**Odd**and**Prime** - Think of numbers that are
**Even**and**Multiples of 5** - Think of numbers that are
**Less than 20**and**Factors of 60** - Think of numbers that are
**Square Numbers**and**Multiples of 3** - Think of numbers that are
**More than 20**and**Triangular Numbers** - Think of numbers that are
**Prime**and**Square Numbers**

3) I think this is a very good activity for partners. I copied half of the heading cards and number cards on a different color of copy paper so that two sets of partners working at the same table would not mix up their cards.

4) It was very easy for the students to accidentally disturb some or all of their pieces which caused some frustration. Next year I plan to enlarge the puzzle and copy on card stock and/or laminate. Several different colors of heading and number cards would also be helpful.

5) When partners believed that they had completed the puzzle correctly, I asked another partnership to look at their puzzle to see if they could find any errors. This on the fly extension helped push the ceiling even higher for early finishers.

5) When partners believed that they had completed the puzzle correctly, I asked another partnership to look at their puzzle to see if they could find any errors. This on the fly extension helped push the ceiling even higher for early finishers.

Here are some photographs of completed puzzles from my classes. There is at least one error on each of these puzzles and I am thinking I could use these photos next year to help introduce the activity and/or use as an extension.

This is a very cool chart that one of my students created on her own to help her logically and methodically determine where heading cards and number cards could or could not be placed in order to solve the puzzle.

And here is a link to some correct solutions to the puzzle posted on the NRICH website.

## Sunday, August 20, 2017

### Student Survey Questions: Middle School Edition

Surveys are a great way for students to practice collecting data and then making sense of that data by create a graph to display the data and finding the measures of central tendency. Here is a list of survey questions that are appropriate and engaging for middle school students. Please share any ideas you have for survey questions and data displays and analysis in the comments!

Here is a link to the document I created for this project. The survey questions below are included in the document for student to select from or get examples in order to write their own survey question.

Data and Statistics Project

1) How many pets do you have?

2) What time did you go to bed last night?

3) How many hours of sleep did you get last night? (round to the nearest 1/2 hour)

4) How many letters are in your first and last name?

5) How many books have you read this school year?

6) How many siblings do you have? (include half-siblings and step brothers and sisters)

7) How many movies did you watch in the theater last month?

8) How many times did you buy your lunch at school last week?

9) What time did you wake up last Saturday morning?

10) How many glasses of milk do you drink daily?

11) What is your shoe size?

12) How many states have you visited?

13) How many texts do you send a day?

14) How many pieces of candy to you eat on Halloween?

15) If I gave you 34 hot wings right now, how many would you eat?

16) How many times a week do you do chores?

17) How many times a week do you eat fast food?

18) How many digits of pi do you know?

19) How many times do you watch TV in a week?

20) How much money do you spend when you go to town?

21) How many pairs of shoes do you have?

22) What is your height?

23) How long does it take you to get ready in the morning? (in minutes)

24) How many times a day do you brush your teeth?

25) How many times a week do you go shopping?

26) How many hours of sports do you play in a week?

27) How many minutes do you spend a night on your homework?

28) What time do you get up in the morning?

29) How many times have you been to a different country?

30) How many people do you consider to be your closest friends?

31) How much money would you need to be given to drink sour milk?

32) How many sports do you play?

33) Flip a water bottle 10 times, how many times does it land upright?

34) How many types of soda do you drink?

35) Write your own

Here is a link to the document I created for this project. The survey questions below are included in the document for student to select from or get examples in order to write their own survey question.

Data and Statistics Project

1) How many pets do you have?

2) What time did you go to bed last night?

3) How many hours of sleep did you get last night? (round to the nearest 1/2 hour)

4) How many letters are in your first and last name?

5) How many books have you read this school year?

6) How many siblings do you have? (include half-siblings and step brothers and sisters)

7) How many movies did you watch in the theater last month?

8) How many times did you buy your lunch at school last week?

9) What time did you wake up last Saturday morning?

10) How many glasses of milk do you drink daily?

11) What is your shoe size?

12) How many states have you visited?

13) How many texts do you send a day?

14) How many pieces of candy to you eat on Halloween?

15) If I gave you 34 hot wings right now, how many would you eat?

16) How many times a week do you do chores?

17) How many times a week do you eat fast food?

18) How many digits of pi do you know?

19) How many times do you watch TV in a week?

20) How much money do you spend when you go to town?

21) How many pairs of shoes do you have?

22) What is your height?

23) How long does it take you to get ready in the morning? (in minutes)

24) How many times a day do you brush your teeth?

25) How many times a week do you go shopping?

26) How many hours of sports do you play in a week?

27) How many minutes do you spend a night on your homework?

28) What time do you get up in the morning?

29) How many times have you been to a different country?

30) How many people do you consider to be your closest friends?

31) How much money would you need to be given to drink sour milk?

32) How many sports do you play?

33) Flip a water bottle 10 times, how many times does it land upright?

34) How many types of soda do you drink?

35) Write your own

I got some ideas for survey questions from f(t) here:

## Sunday, May 21, 2017

### Area of Polygons Discovery Lesson on Dot Paper

I created these pictures to give my students the opportunity to explore the areas of triangles, parallelograms and trapezoids in reference to the area of rectangles. I made copies of these two slides from my lesson presentation for the students to write on and they explored the areas of the shapes by drawing on the dot paper, counting square units, cutting off parts of the polygons and moving them to different places, and making conjectures about how the areas of the figures were related and why. This was a very engaging activity and the students used many different strategies to find the areas of these figures and these explorations formed the foundation for understanding the formulas for finding the areas of these figures. I was very pleased to see that students continued to use ideas from these pictorial models to help them with areas of polygons that were not displayed on dot paper (for example, triangle from left side of parallelogram is moved to the right side to form a rectangle with the same area).

The students had lots of interesting ideas for finding the area of Triangle F. I was amazed and thrilled when several students, connected the area of the obtuse triangle to the area of a paralleogram, instead of a rectangle, which was much easier for students in the class to see.

I use this slide to summarize the exploration. I highlighted part of each of the formulas in red to help students see the connection between all of the formulas: base x height. Even the formula for trapezoid can be understood in this way if we talk about finding the average base x height.

## Thursday, February 9, 2017

### Cliff Jumping and Absolute Value

I love to show this video of Laso Schaller's world record jump of 58.8 meters in Maggia, Switzerland.

I use it when we start our unit on absolute value and it sparks a nice discussion about the distance above and below zero. I show the video and then ask the students what they would want to know before they jumped off the cliff. They usually come up with some very interesting comments including how deep the water is and how far down you will go after you hit the water. Using these questions I guide the discussion towards the total distance traveled by drawing a diagram on the board with a vertical number line showing positive numbers above zero and negative numbers below zero. I use 192 ft (58.8 m) for the top of the cliff and -48 ft. for the depth of the water needed for the jump. I got these numbers by using the following information about the depth of water needed for a safe jump based on the height of the jump.

This discussion helps provide the mathematical motivation for absolute value and why positive numbers remain positive and negative numbers become positive when we are measuring distance. And then at this point I offer the mathematical notation and definition of absolute value.

Here are some still shots from the video:

The video replays the jump several times and shows different angles and information with each replay. For some of the replays, additional information about the jump overlays the video including the height of the top of the cliff as it is compared to the Leaning Tower of Pisa and also a time lapse of the speed and time of the Laso Schaller throughout the jump.

I use it when we start our unit on absolute value and it sparks a nice discussion about the distance above and below zero. I show the video and then ask the students what they would want to know before they jumped off the cliff. They usually come up with some very interesting comments including how deep the water is and how far down you will go after you hit the water. Using these questions I guide the discussion towards the total distance traveled by drawing a diagram on the board with a vertical number line showing positive numbers above zero and negative numbers below zero. I use 192 ft (58.8 m) for the top of the cliff and -48 ft. for the depth of the water needed for the jump. I got these numbers by using the following information about the depth of water needed for a safe jump based on the height of the jump.

(Depending on time constraints, I could leave off the numeric expression and ask the students to figure out what the depth of the water must be for this jump.)

Here are some still shots from the video:

The video replays the jump several times and shows different angles and information with each replay. For some of the replays, additional information about the jump overlays the video including the height of the top of the cliff as it is compared to the Leaning Tower of Pisa and also a time lapse of the speed and time of the Laso Schaller throughout the jump.

## Sunday, January 29, 2017

### A good problem is worth a thousand worksheets

I got this problem from openmiddle.com.

Low floor high ceiling tasks are also sometimes called low threshhold high ceiling or low entry high ceiling tasks. NRICH describes a low threshhold high ceiling task like this:

"A LTHC mathematical activity is one which pretty well everyone in the group can begin, and then work on at their own level of engagement, but which has lots of possibilities for the participatns to do much more challenging mathematics."

I am looking forward to trying this problem with my students now that we are working on ratios:

## Saturday, December 31, 2016

### 3D Shapes Exploration

I like to set up an exploration of 3-D shapes at the beginning of the unit on volume and surface area. Last spring I set up six stations using the outdoor courtyard of our school and the students enjoyed hands on explorations of 3-D shapes in the sunshine. It was wonderful.

1) Cutting, Coloring and Folding Paper Nets of 3-D Shapes

2) Building 3-D Shapes with Magnetix

3) Building 3-D Shapes with Zome Tooels (and bubbles!)

4) Drawing Nets and 3-D Shapes with Sidewalk Chalk

5) Building 3-D Shapes with Toothpicks and Marshmallows

6) Exploring with 3-D Models and Removable Nets

I also created this graphic organizer that I ask the students to fill out as they work at the stations and explore six 3-D shapes. I ask the students to focus on filling out the first six columns of the graphic organizer during the eploration and then reference it during subsequent lessons on volume and surface area. This is a good introduction to the vocubulary of three dimensional solids. Graphic organizer can be downloaded here.

1) Cutting, Coloring and Folding Paper Nets of 3-D Shapes

2) Building 3-D Shapes with Magnetix

3) Building 3-D Shapes with Zome Tooels (and bubbles!)

4) Drawing Nets and 3-D Shapes with Sidewalk Chalk

5) Building 3-D Shapes with Toothpicks and Marshmallows

6) Exploring with 3-D Models and Removable Nets

I also created this graphic organizer that I ask the students to fill out as they work at the stations and explore six 3-D shapes. I ask the students to focus on filling out the first six columns of the graphic organizer during the eploration and then reference it during subsequent lessons on volume and surface area. This is a good introduction to the vocubulary of three dimensional solids. Graphic organizer can be downloaded here.

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