Array Multiplication Cards

Even though this is a blog that dedicates most of the resources to grades 7-12, sometimes we have to have some help with the basics for those students. In Ontario we have a new initiative called Focus on the Fundamentals and even though you could argue that we haven't forgot the fundamentals, perhaps an the idea of putting a little extra attention on the fundamentals may not be a bad idea. In this case we are looking at students "knowing" their multiplication facts. Knowing is in quotations because what one person thinks of as knowing may not agree with others. For example, memorizing the multiplication tables doesn't necessarily mean that a student knows what multiplication is.

Cut to a couple of weeks ago. We was in a primary math session from @gfletchy. In that session, we used these 10 frame cards that were basically a game to help students recognize numbers. It seemed like an engaging way to do that. With a little bit of searching, we found that he also has cards for multiplication that focus on groupings and go up to 7x7. Since we thought the idea of practicing multiplication tables would be good for grade 7&8 students, we thought a more advanced representation might be as arrays. So here are cards that can be used to practice multiplication facts up to 12x12.

Right now there are two versions and two sets of each version. We have one version with just the dots and one set with the dots with rectangles around groups to highlight the arrays a bit more. Each version also has two sets, one with the answers on the back (for kids to work in pairs) and one without (for group play).

To help with the counting of the dots on each side we have put vertical and horizontal lines to mark groups of 5 dots. This way students can see, for example, there is a group of 5 and three more on one side and two groups of 5 and 2 more on the other side so this must be 8x12. The lines can also help by letting students use decomposition to break the problem up into smaller simpler problems which they can add together. This is an effective strategy to use on their way to internalizing the multiplication table. One thing you might want to do is show them one card and just ask them to Notice and Wonder about what they see and hopefully they can recognize what the lines indicate.

• All Grades - As review 

1. Print out the cards on card stock, cut and laminate them. You will probably want more than one set. We recommend printing each set out using a different colour of card stock. This way if the sets get mixed up then you just need to match the colours.
2. When you print out the cards, the first 2 pages cover 2x1 all the way to 7x7 (with a few times one cards in there to fill the page). The next page cover 8x2 all the way to 9x9. And finally the last two pages go from 10x2 all the way to 12x12. So if you have kids struggling still with multiplication, you may want to limit them to some of the first few pages.
3. If you are printing out the cards with the answers on them, the answer pages show up every second page with the intent that when you print them, you have double sided checked off on your printer/copier. If you have the option, have it "Flip on the long edge". 

1. For Game Mode: Put kids in groups of 3-6.
2. Shuffle the cards (versions without answers on the back)
3. Someone flips over a card. 
4. The first person to say the correct product gets the card (or a point). Students have to agree that that is the correct answer.  If a student says more than one answer, they are disqualified for that card. 

You might be concerned that speed of calculations may come into play here and if you play in Game Mode, you wouldn't be wrong. One way for speed to be a factor is to start out with just the easiest of cards (first two pages) and only move on when the majority of students have mastered them. Or you could have students write their answers on white boards and then not reveal the answer until everyone has completed (with not worry about points or who answered first). 

1. Conversely, you could have kids work in pairs and use the decks with the answers on the back. 
2. Shuffle the cards (version with the numbers on the back)
3. Deal out half to each person. 
4. Each student takes their deck and holds them so the other can't see either side. 
5. They take turns showing each other a dot array and listen for their partner to say the answer (visible on the back). 

• Array Multiplication Cards (Google Doc, PDF)
• Array Multiplication Cards with answers (Google Doc, PDF)
• Array Multiplication Cards (Groups) (Google Doc, PDF)
• Array Multiplication Cards (Groups) with answers (Google Doc, PDF)
• Google Drawing Templates for each card (UnGrouped, Grouped)

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

Sort Students into Groups using Percents, Fractions and Decimals

In this activity students are each given one card. The card will either have a fraction, percent or decimal. Their job is to find the two other students who have the same value but a different representation. This shouldn't take too long and could be repeated every couple of days just to solidify conversion between fraction, decimal and percent.

If you wish you can also have students do this individually with this Desmos cardsort.

• Gr7 - determine, through investigation, the relationships among fractions, decimals, percents, and ratios
• Gr8 - translate between equivalent forms of a number (i.e., decimals, fractions, percents)
• MPM1D, MFM1P - As review

• Download the cards and cut them out (you may want to put them on cardstock and laminate)

1. Shuffle the cards and distribute one per student. Note that there are 12 sets of 3 cards so you may want to remove sets to more closely match your student population. 
2. Instruct students to find the two other people that have the same value but a different representation. 
3. Once students find their partners they will be in groups of three,

• Group Fractions, Decimals, Percent Cards (Googledoc) (pdf)
• Individual Desmos Cardsort Version

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

Fraction Operation Continuum

As math teachers we definitely want our students to practice to become proficient but piles of problems or worksheets aren't going to be very engaging to students. We think this tweak to the standard worksheet is a way to turn those boring questions into something more engaging.

In Ontario, grade 7s are introduced to operations with fractions. Addition & subtraction and multiplication & division with whole numbers. The premiss here is fairly simple. Students are presented with multiple cards of questions (in this case of adding and subtraction of fractions (with a little of division and multiplication with whole numbers). The cards represent problems that increase in difficulty as you go from one to the next. Students can all start at the first envelope or you could give them an exit card the day before to help place them in a particular card to start. Students check their own answers using answer cards with the answers written with "invisible" ink that can be revealed by shining a UV light on it.

Students really seem to like this style of activity as they feel empowered to move from card to card when they are ready and the added feature of checking the answers with the UV pen gives a sense of novelty. This could be used as practice or review.

• Gr7 - divide whole numbers by simple fractions
• Gr7 - use a variety of mental strategies to solve problems involving the addition and subtraction of fractions
• Gr7 - add and subtract fractions with simple like and unlike denominators, using a variety of tools (e.g., fraction circles, Cuisenaire rods, drawings, calculators) and algorithms;
• Gr7 -  demonstrate, using concrete materials, the relationship between the repeated addition of fractions and the multiplication of that fraction by a whole number
• Gr8 - As review (we plan an extension so that this could be used for grade 8 with multiplying and dividing fractions)

• Enough copies of each of the question cards for your class (there are four cards per page at each level) in different colour card stock for each level,  laminated (use colours that allow seeing the magic pen writing - you may want to test this). You will likely not need as many cards in the last few envelopes as students work at different paces. You will need as many as you have in your class if you decide to start everyone at the first level. Fewer if you let students start at different levels (see below)
• 3 sets of the answer cards (use magic pen to write the answers anywhere along each equation, they could be sideways, upside-down, (the answers are on the last page of the Google Doc). The answer cards are the same as the question cards but you write the answers in invisible ink on them. To help distinguish the answer cards to the question cards you should put a stamp or sticker on the back. Write on the cards first then laminate them. If you write on the card after lamination then the ink tends to wear off. There is a separate answer card on the last page of the download. That is for you to carry around (or not) but not for showing students but more for your reference.
• 3 "magic" pens can be purchased at Chapters/Indigo or we found these at a Scholastic's book fair. We have since purchased some on eBay or Amazon.

1. Place the questions in piles (or in envelopes taped to the wall) in order of difficulty and set up three stations for the answer cards. Students will get a card and answer the first 5 questions. 
   
2. You could have all students start at level 1 but for this activity to be most successful, students should start at the appropriate envelope. If they start in one that is too hard they will be frustrated and if they start in one that is too easy they will be bored. Use an exit card (the day before) to help you decide which envelope each student should start in. When given back the exit card write down the level they will start in. 
3. To check their answers, they will go to a station and use the magic pens. Students may decide to do one question at a time and then go check their answer or they may do all 5 and then check. Students are monitoring themselves so they decide. If they get the first 5 right, they have a level of mastery to move themselves to the next card. If not there are more questions on the card until they master that type. You can decide whether you want them to do the other 5 or just do enough to get a total of five correct. 
4. As they move through the continuum, the hope is that they reach level 6 which matches the grade 7 curriculum. Since our goal is to get them to level 6, students should solve ALL equations on that card instead of just five. 
5. The seventh level is set up to challenge students who are moving forward quickly. They should solve all questions on this card.  

• Fraction Operation Continuum (with Answers) - (PDF, Google Doc)

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

Tarsia Puzzle - Fractions,Decimals and Percents

We came across these puzzles a few years ago. The premise is that you have a bunch of questions and matching answers. Students have cards that have either an answer or question on the edge. They then have to match up the edges that have the pairs of questions and answers and eventually it will make a shape. In this case the activity is matching up different representations of the same number (Decimals, Percents and Fractions).
We found these puzzles originally on the Mr Barton Maths site under the Tarsia Jigsaw Page. The nice thing here is the Tarsia software can be downloaded here (click the green button to download - unfortunately it is Windows only) and you can edit or create your own puzzles. There are even banks of hundreds of already created puzzles on the site (just scroll down on the Tarsia page and look for the smiley faces for zipped files). So this puzzle was one of theirs that we tweaked a bit. These are great puzzles to pull out when you want to break up a long class or have 10 min at the beginning or end of a class to fill. 

• Gr7 - determine, through investigation, the relationships among fractions, decimals, percents, and ratios
• Gr8 - translate between equivalent forms of a number (i.e., decimals, fractions, percents)
• MPM1D, MFM1P - As review

Tarsia Card Prep work: There are two downloads (three if you include the actual Tarsia file). The actual cards for the puzzle and the answer card. The cards for the puzzle come on 2-4 sheets (there are different styles of puzzles) and each triangle needs to be cut out (see image to the right). We have found that if you have several sets of the same puzzle (say if students do it in groups of 3) then it is best to copy each set in a different colour. That way it is harder to get the sets mixed up and easier for clean up after (ie you only need to check if there are 18 cards of each colour). We find that the cards last longer if you laminate them first then cut them.

1. This activity can be done in groups or individually and can take about 15 min depending on the student.
2. Hand students the entire set and ask them to match up the different representations of the same number.  The outside edges are blank.
3. In this case the finished puzzle looks like a hexagon.

• Note: this puzzle is challenging because many of the numbers appear more than once (eg 75% shows up as a decimal twice and a fraction three times)

The video, below, is only visible in the WECDSB domain. That is, only teachers in our school board can see the video if they are logged into their MyTools2Go accounts

• Fractions, decimals and perccent Tarsia cards (pdf)
• Fractions, decimals and percent Tarsia solution (pdf)
• Fractions, decimals and percent Editable (xjsw)

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 - Perfect Square Practice

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). One of the advantages of doing this is that the bulk of the time spent on the software is actually doing math rather than building something. 

In this sketch students can practice recognizing perfect squares up to 144. It is a very simple sketch not meant to take much time but to just familiarize students with the first 12 perfect squares as well as to remind them that perfect squares can also be defined by physical squares.

• Gr7NS1.6 - represent perfect squares and square roots, using a variety of tools (e.g., geoboards, connecting cubes, grid paper);
• Gr8 - could be used as review or see our square root guesser sketch instead

• 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

This sketch randomly selects a number under 150 and asks students whether it is a perfect square. They can make a mental guess and check their answer. Or, before the check their answer,  if they want to test it out they can try to create a square that has area equal to the given number. Once done they can generate another random number and try again. The hope is that this will help them become familiar with the first 12 perfect squares. Watch this video to see a demonstration of how it works.

• PerfectSquares.gsp (iPad/V5)
• Web sketch here
• For more sketches like this go to my full page

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 - Square Root Number Line Guesser

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). One of the advantages of doing this is that the bulk of the time spent on the software is actually doing math rather than building something. In this sketch students can practice their knowledge of estimating the square root of numbers up to 500. There are several levels of difficulty: perfect squares up to 100, perfect squares up to 500, square roots up to 100 and square roots up to 500. The intent was that this was built as a practice file for grade 8 students but grade 7 students could use it to practice perfect squares.

• Gr7NS - represent perfect squares and square roots, using a variety of tools
• Gr8NS - estimate, and verify using a calculator, the positive square roots of whole numbers, and distinguish between whole numbers that have whole-number square roots (i.e., perfect square numbers) and those that do not

• 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 the sketch works

• SquareRootNumberlineGuesser.gsp (iPad/V5)
• Web sketch here
• For more sketches like this go to my full page

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