Introducing Histograms

In grade 8 in Ontario histograms are one of the new data management topics. Below is a series of activities that range from paper & pencil to virtual manipulatives. They start by a simple comparison of bar graphs and histograms to identify characteristics, then moves to some consolidation of ideas using a Desmos Sort. Then some data collection of student heights to formally introduce creating a histogram followed by some practice making histograms by hand and then with Google Sheets. If you did everything here it should take about 2-3 classes.

Grade 8

• collect and organize categorical, discrete, or continuous primary data and secondary data (e.g., electronic data from websites such as E-Stat or Census At Schools), and display the data in charts, tables, and graphs (including histograms and scatter plots) that have appropriate titles, labels (e.g., appropriate units marked on the axes), and scales (e.g., with appropriate increments) that suit the range and distribution of the data, using a variety of tools (e.g., graph paper, spreadsheets, dynamic statistical software); 
• select an appropriate type of graph to represent a set of data, graph the data using technology, and justify the choice of graph (i.e., from types of graphs already studied, including histograms and scatter plots);
  
• read, interpret, and draw conclusions from primary data (e.g., survey results, measurements, observations) and from secondary data (e.g., election data or temperature data from the newspaper, data from the Internet about lifestyles), presented in charts, tables, and graphs (including frequency tables with intervals, histograms, and scatter plots);
• demonstrate an understanding of the appropriate uses of bar graphs and histograms by comparing their characteristics
• Could also be used as a review in MBF3C or MDM4U

• If you are doing the physical card sort then print graphs to card stock (we suggest each set is a different colour so that if students mix them up they are easy to separate). We also suggest lamination. There are two pages for a total of 16 cards. 
• If you are doing any of the online activities then chromebooks/laptops/computers/iPads will be needed. You will also need to make copies of the Desmos version of the card sort and/or the Desmos card sort consolidation.
• Copies of the handout. 

1. The purpose of the card sort is to start to distinguish both histograms and bar graphs and continuous vs categorical data. But first have them do an open sort. Hand out cards to students (or give students the code for the Desmos Sort). Ask them to sort them in any way they wish. The only stipulations are that there should be at least two groups and each group must have at least two cards. Students will sort them in all kids of ways (by the numbers, by the topics, by the looks). Circulate and encourage them to explain how they were sorted.
2. Once sorted it is likely that most will not have them sorted how you wish. Have them describe their sorts then ask them to sort them in a way so that there are only two groups. 
3. By this time some may have them sorted into bar graphs and histograms. If not show one bar graph and then one histogram stating that you want those cards to represent characteristics of each group (you can use the slideshow to show the graphs). Use these graphs to develop the difference between continuous and discrete data.
   
4. Time to change the pace and have students collect some data. Have students measure their heights and put them on a dot plot on the board. Use this to create a histogram (without creating a tally). You might want to collect their heights in a spreadsheet so that you can create a histogram with it later. 
5. Walk through the front of the handout to show how to create a histogram from data. 
6. Likely you will be at least done one class at this point (if not more). At the beginning of the second class, start with the consolidation Desmos card sort (with some stuff on average) to remind students of continuous and discrete data. Another possible way to start is to have students use this Google form to enter some examples of continuous and discrete data. You can take up the results. 
7. Once done taking up the worksheet you can then introduce students to creating histograms using Google Sheets. Share this sheet with continuous data. This video will tell you how. 

• Card Sort Google Docs, PDF, Desmos Slide show of each graph
• Desmos Consolidation
• Handout Google Docs, PDF 
• Handout Answers Google Docs, PDF
• Discrete/Continuous Form
• Spreadsheet Graphs, Answers

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks 

Geometer's Sketchpad - Practice Line of Best Fit

When using the Geometer's Sketchpad it is often better to "start from sketch, not from scratch". That is, give students a premade sketch rather having them build something from nothing (as many textbooks would have you do).
In this activity, students practice placing the line of best fit on a linear set of data. It's not meant to be really difficult but just to reenforce the idea of what the line of best fit is. Students can check their answer and try as many as they like. Clicking the Medium or Hard buttons will spread the points out more randomly to make the line a bit harder to determine. This is not meant to be really hard but just a quick way to determine if students have the basic concept of what a line of best fit is

• MPM1D, MFM1P - construct tables of values, scatter plots, and lines or curves of best fit as appropriate, using a variety of tools (e.g., spreadsheets, graphing software, graphing calculators, paper and pencil), for linearly related and non-linearly related data collected from a variety of sources
• MAP4C - D1.4 - create a graphical summary of two-variable data using a scatter plot (e.g., by identifying and justifying the dependent and independent variables; by drawing the line of best fit, when appropriate), with and without technology
• MDM4U - D2.4 - generate, using technology, the relevant graphical summaries of two-variable data (e.g., scatter plots, side-by-side boxplots) based on the type of data provided (e.g., categorical, ordinal, quantitative)

• All that is needed is the electronic download (below)
• Note that this really works well on an iPad using the Sketchpad Explorer App (which is free)
• You can also use this on any web based computer (or Chromebook) with this Web sketch

Watch the video below to see how to use the sketch

• Line of Best Fit.gsp (iPad/V5)
• Web sketch here
• For more sketches like this go to my look at the dynamic web sketch tab above

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

The 12 Days of Pascal's Triangle

This is a simple paper & pencil activity that is supposed to be a light exercise to bring in the Christmas season. We got the original idea for this from this site and modified it a bit. It is not particularly taxing but could be used when talking about Pascal's triangle or sequences and series in general (and of course if it's Christmas time). As it is the assignment is fairly simple but it could easily be extended to have students find the general term of both the series and sequences.

• MCR3U -  C1.5 determine, through investigation, recursive patterns in the Fibonacci sequence, in related sequences, and in Pascal’s triangle, and represent the patterns in a variety of ways; C2.2 determine the formula for the general term of an arithmetic sequence or geometric sequence, through investigation using a variety of tools and strategies, and apply the formula to calculate any term in a sequence; C2.2 determine the formula for the sum of an arithmetic or geometric series, through investigation using a variety of tools and strategies, and apply the formula to calculate the sum of a given number of consecutive terms 
• MDM4U - A2.4 make connections, through investigation, between combinations and Pascal’s triangle

• Just the hand out

• Hand out the sheet and let the Pascal/Christmas joy begin

• 12 Days of Pascal (with answers) (doc) (pdf)

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

How do I compare to Michael Phelps?

When talking about linear relationships and scatter plots, a common activity is to have students measure their various body parts and then compare sets of them to see if there is a correlation. This activity builds on an activity in the TIPS package for grade 9 Applied (Section 3.1.2-4 pg 4). Kids usually like that activity (probably since they get to move around). This is a small tweak to make that activity a little more engaging. In this activity, students compare their arm span, foot size and hand size to some unknown people (arm span is Michael Phelps, foot size is Shaquille O'Neal, hand size is Michael Jordan - yes some still know who he is) and the heights of the tallest and shortest humans (among others).

• MFM1P, MPM1D – B1.4 describe trends and relationships observe in data, make inferences from data, compare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses
• MDM4U - D2.3 generate, using technology, the relevant graphical summaries of two-variable data (e.g., scatter plots, side-by-side boxplots) based on the type of data provided (e.g., categorical, ordinal, quantitative)
• MAP4C - D1.3 generate, using technology, the relevant graphical summaries of two-variable data (e.g., scatter plots, side-by-side boxplots) based on the type of data provided (e.g., categorical, ordinal, quantitative); D1.4 generate, using technology, the relevant graphical summaries of two-variable data (e.g., scatter plots, side-by-side boxplots) based on the type of data provided (e.g., categorical, ordinal, quantitative) 

The material set up for this one is a bit labour intensive. There are some that require taping multiple sheets together

• Michael Phelps - this is probably the hardest to pull off. We took this image of Phelps and wanted to blow it up so that it's arm span matches his actual arm span of 6'7" (201cm). To do this we changed the contrast so that the face was less recognizable then used the Poster Razer to slice the image up into 8 pieces (this is actually a free program that is great for making large posters out of smaller pieces of paper). We then printed them on 31"x23" poster sheets and then taped them together and laminated them.

• Shaquille O'Neal - Shaq's feet come in two pieces that have to be taped together
• Michael Jordan - No special instructions for the hands
• Height Wall - We have provided for you some samples to include on the height wall, including some really tall and really short ones and some popular culture examples. We encourage you to add your own examples (ones that resonate with your own students). Cut these out to be placed on the height wall for students to see and compare to.
• Other materials that you will need are measuring devices (metre sticks, measuring tapes etc) 

1. Prior to students arriving in class, tape the feet to the floor at the classroom entrance, tape the hands on the wall where they will be visible, tape the arm span on the wall so there is room for kids to measure up to it and find a spot to make your height wall. The height wall should have a measuring tape on it (or metre sticks) so students can measure themselves but also stick the sample heights of the famous people on the wall to have students compare to them.
2. As a Minds On you might want to show this image and ask "How tall would the person be to fit this shoe?"
3. Students start by completing p. 4 from TIPS.
4. Students record their measurements on p. 5 and as they circulate they can compare their own measurements to those on the walls and floor. Along the way they can make guesses as to who the mystery feet, hands and arm span are.  
5. Collect students' data
6. Ask students to reveal their guesses.  Have them discuss why they chose these people.
7. Invite students to add there own person to the height wall.
8. As an extension, you can collect this data in a spreadsheet or online form. We have created a sample Google form (or use this link http://bit.ly/relationshipform). If you want your own copy, contact us and we will share one with you. Here is a link to the spreadsheet of data. When you use this form be sure to have students put in a unique class identifier so that you can find your class data in our spreadsheet of data.  
9. You can analyse the data with students using Fathom. (What is the best predictor of your height? Is your foot length the same as your forearm length?)
10. Some extension material on Michael Phelps can be found in this Smart Notebook file and this documentary called Miracle Body.

• Scans of the feet and hands (pdf) - note Phelps is light here and you can recognize him
• Scans of arm span posterized (pdf)  - this is a darker scan of just the arm span pieces.
• Michael Phelps original image (Dark) (Light)
• Michael Phelps info File (not) (pdf)
• Height Wall (doc) (pdf)
• TIPS Activity (doc) (pdf)

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

Data Management Scavenger Hunt

This is an activity that we adapted from a TIPS activity (Grade 8 Unit 11, Day1, Pg 3). In this activity student are given a set of data (with no labels) and asked to find their matching graph and title (which are randomly placed around the room). Once finding their cards they are given a question card to use with their graph and data. It's actually a surprisingly difficult activity but it helps bring out conversations about types of data and what they should look like.

• Gr7DM - select an appropriate type of graph to represent a set of data, graph the data using technology, and justify the choice of graph; read, interpret, and draw conclusions from primary data and from secondary data presented in charts, tables, and graphs (including relative frequency tables and circle graphs);
• Gr8DM - select an appropriate type of graph to represent a set of data, graph the data using technology, and justify the choice of graph; read, interpret, and draw conclusions from primary data and from secondary data, presented in charts, tables, and graphs (including frequency tables with intervals, histograms, and scatter plots);
• MBF3C - D1.5 identify different types of one-variable data, and represent the data, with and without technology, in appropriate graphical forms (e.g.,histograms, bar graphs, circle graphs, pictographs)
• MAP4C - D1.4 create a graphical summary of two-variable data using a scatter plot , with and without technology
• MDM4U - D1.5 interpret statistical summaries to describe the characteristics of a one-variable data set (note that this certainly can be used as review of all types of data)

Data Management Cards Prep work: Print, laminate (optional) and cut the cards out. Separated out the sets of data from the graphs and Titles. Each page contains one set of data, graph and title (note that one set has two possible matching graphs). Also included in the set are a page with extra questions and a page with the answer key on it.

1. Prep the classroom by randomly sticking the graphs and the titles around the room for students to find. Note that this activity was originally made for grade 8 but if you wanted to do it for grade 7 you should probably consider taking out any graphs with scatter plots.
2. Put students into groups of 2 or three (there are 17 different sets of cards)
3. Each group gets a set of data and are told to wander the room looking for their matching graph and title. Some of these are not easy so students should work together to decipher the information. You should probably circulated to give hints (with your answer key).
4. Once a group finds their matching cards, they are given a question card to answer regarding their graph.
5. If they finish early give them another set of data to try.

Data Management Scavenger Hunt (Doc) (PDF)

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

Sort Graphs into Types of Data

This is a simple and quick sorting activity for graphs to help introduce the idea of types of data. And although that is the intended purpose it is done as an open sort so that students can (hopefully) start to see that there are different types of data without being formally introduced to them. This is not intended to be a long activity but just a short Minds On type to begin a class. Keep in mind that if students don't sort them by data type then you may have to ask some leading questions to help move them that way. Though this was originally developed for the MDM4U class, by removing some of the graphs, it could be used for grades as low as 7 (for example, grade 7 students will likely not have seen histograms).
[Updtae - Now a Desmos Card Sort version exists]

• Gr7DM, Gr8DM - collect and organize categorical, discrete, or continuous primary data and secondary data
• MBF3C - identify different types of one-variable data (i.e., categorical, discrete, continuous)
• MDM4U - distinguish different types of statistical data (i.e., discrete from continuous, qualitative from quantitative, categorical from numerical, nominal from ordinal, primary from secondary, experimental from observational, microdata from aggregate data)

• One set of graph cards per group

1. put students in groups
2. distribute cards with graphs 
3. ask them to sort in any way they see fit. Use the results to steer towards types of data. Students should focus on both the look of the graph and the characteristics of the data being shown.
4. possible leading questions: "Can you arrange these so that there are only two groups? If so can you arrange each of those two groups into sub groups?" 
5. As an extension or alternate activity, students could sort by types of graph.

Note that the graph of the number of friends is a histogram which might lead some to think it is continuous data. This could be a good talking point as the number of friends a person can have can only be a discrete amount. 

Types of Data cards (doc) (pdf)
Desmos Card Sort (with self checking feature for numerical and categorical)

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks