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

Tarsia Puzzle - Multiplying and Dividing by powers of 10

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.
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. So this puzzle was one of our creations. 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). 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.

• Gr8NS - multiply and divide decimal numbers by various powers of ten

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 tot 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 questions have the same digits.

• Multiplying and Dividing decimals by powers of 10 Tarsia cards (pdf)
• Multiplying and Dividing decimals by powers of 10 Tarsia solution (pdf)
• Multiplying and Dividing decimals by powers of 10 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

Integer Multiplication Mind Reader

This one is pretty simple but the students seem to really like it. We got the idea from a colleague and am sure we aren't the first to do it. Basically you get groups of three. Two students grab a card from a deck and without looking at them put them on their foreheads facing out. The third student multiplies the two numbers and states the product. Those holding the cards then try to guess the two numbers.

• Gr8NS - represent the multiplication and division of integers, using a variety of tools 
• MPM1D - As review
• MFM1P - As review

• All that is needed are decks of cards. Here black cards are positive and red cards are negative (or vise versa if you prefer). You may wish to remove the face cards or come up with protocol for which numbers the face cards represent. On side note, if you use the face cards, don't assume that all students will know what a face card is (ie a card that literally has a face on it) as it seems that more and more students may not have the experience of playing cards at home. 
• Optional materials might be a personal portable white board and dry erase marker for students.

1. Create groups of three students. 
2. In the group choose 2 mind readers and one product leader
3. Each mind reader chooses a card and without looking at either card, place them face out on their foreheads. 
4. The product leader multiplies the two integers (red is negative, black is positive) and states the product (or writes it on their whiteboard)
5. The mind readers then take turns guessing the factors (cards) they are holding).
6. The first to guess correctly becomes the new product leader and the original product leader becomes a mind reader. 
7. Repeat as often as you wish

• Integer Mind Reader Instructions (pdf)

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