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

Rate of Change Continuum

A continuum is something where the level difficulty increases in incremental steps. In this case the continuum is dealing with calculating the rate of change (slope) of a linear relationship.
We have previously posted a continuum for solving equations here and here but this one is a bit different. This one has five levels of determining the rate of change from a graph (in context) for a linear relationship. The first level shows lattice points, a rate triangle and the calculation of both rise and run (super basic) and the difficulty increases with each level (see below) until the last level where there is only a scale with no grid lines (so the answer is more of an estimate).

Each page has 6 graphs and students (once they choose the level to start with) choose to answer any three. If they do so correctly then they can move to the next level. The To make things a bit more fun, rather than check the answers with you, we suggest using a UV pen and ink written on the question cards for students to check.
This activity is probably best meant as a consolidation. Note that the expectation is about investigating so hopefully students will have had a chance to develop their own strategies for determining the rate of change. This activity just helps to scaffold it a bit in case they are having trouble (Eg a common mistake that students make when determining the rate of change when the line is in context is to just count boxes for the rise and run without considering the scale).
Note that we also have an Explain Everything version if you have students who have iPads (you may even want to try out the new Explain Everything Collaborative Whiteboard app to have students work in groups from different devices).

• MPM1D, MFM1P - determine, through investigation, connections among the representations of a constant rate of change of a linear relation.

• 20 copies of each of the question cards in different colour cardstock for each level,  laminated (use colours that allow seeing the magic pen writing). Note that you may not need 20 copies of each. Perhaps fewer of the first couple levels and last level as most kids will probably be starting in the 2nd or 3rd level
• 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). To help distinguish the answer cards to the question cards you should put a stamp or sticker on the back.
• 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.

1. For this activity to be successful, students must 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. 
2. Place the questions in piles in order of difficulty and set up three stations for the answer cards. Students will get a card and answer any 3 questions. 
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 3 and then check. Students are monitoring themselves so they decide. If they get the first 3 right, they have a level of mastery to move themselves to the next level. If not there are more questions on the card until they master that type. 
4. As they move through the continuum, the hope is that they reach level 4 which matches the grade 9 curriculum. Since our goal is to get them to level 4, students should solve ALL equations on that card instead of just three. 
5. The fifth level is set up to challenge students who are moving forward quickly. They should solve all questions on this card. They require some estimation and so answers that students get should be approximate. 

Note that for the Explain Everything version, there are still 6 possible graphs for each level but only two on each page. And to check the answer, slide the black ellipse to either the bottom left or right corner. 

• Graphical Rate Continuum (Google Doc, PDF, Explain Everything)
• Entire Folder

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

Number Sentences Sort (update)

One of the smaller expectations we have to deal with is the ability for students to interpret algebraic equations. You know: "what does 2x + 1 mean"?. We created this very simple sorting activity where students are given expressions (and equations) and the sentence to describe them and have to match them up. This is meant to be an activity that is relatively quick. We have two versions here. One for grade 7 that only has expressions and one for grade 8 that has equations as well. We also have an Explain Everything version of each so that if you have an iPad (or Chromebook), with that app, you can have your students sort them electronically. This can also be used as review in Grade 9.
Note: This is an update to the same activity posted last year but now with a grade 8 and Explain Everything version
Double Note: This has been updated again to now include a Desmos card sort. So both card sorts are now transferred to this new Desmos feature. You can learn about Desmos Card Sorts by clicking here. Download the Teacher versions (which you can copy) of these activities below in the download section.

• Gr7PA - translate phrases describing simple mathematical relationships into algebraic expressions using concrete materials
• Gr8PA - translate statements describing mathematical relationships into algebraic expressions and equations
• MPM1D, MFM1P - As review

• For the grade 7 version there are four different (but similar) sets. One set per page. For the grade 8 version there are three different (but similar) sets. One set per page. 
• Print each page on card stock (we also suggest laminating). We suggest that each set be printed on different colour card stock for easy sorting. Cut each out and put each set in an envelope.
• Obviously you will have to decide how many sets you will need for your class depending on whether you pair students up or not. 
• Note that in the version with equations, there are some algebraic expressions that do not have matching sentences. In these cases, students will have to write their own.
• If you choose to use the Explain Everything version, then you probably want to download that .xpl file and put it on a server where your students can get easy access to it. 

Explain Everything Screenshot

1. Depending on how many students you have you may want to do this individually, in pairs or in larger groups. The activity is not super complex so we don't recommend anything bigger than pairs. 
2. Students take each set and sort the algebraic expression with the written version. 
3. When they are done their set they can trade with another group that has a different colour of cards. If they are using the Explain Everything version then they can just go to the next slide. 
4. There is a homework sheet for consolidation that includes both expressions and equations as well.

• NumberSentences-Expressions (doc. PDF, XPL, Desmos)
• NumberSentences-Equations (doc, PDF, XPL, Desmos)
• NumberSentencesHomework-Equations (doc, PDF)
• Entire Folder

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

Row Games

We saw this activity in 2010 when I first found @K8Nowak's blog f(t). I don't know if I would call a Row Game a particularly engaging activity but I am convinced that any way we can make doing boring homework questions more palatable for students is a good thing. The premiss is that you pair students up and they get a worksheet of questions. The questions are in two columns. Each person does one column and if they have done things correctly then their questions on the same row should have the same answer. If they don't then either one or both of them are incorrect and they have to work together to get the correct answer. So this is a self checking activity. We made a bunch of them at the time and I just stumbled upon them this week so we thought we would post them. These ones are for ratios, proportion, simplifying expressions and solving simple equations.

MFM1P, MPM1D

• illustrate equivalent ratios, using a variety of tools
• solve for the unknown value in a proportion, using a variety of methods 
• make comparisons using unit rates
• solve problems involving ratios, rates, and directly proportional relationships in various contexts, using a variety of methods
• solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms
• add and subtract polynomials involving the same variable up to degree three, using a variety of tools
• multiply a polynomial by a monomial involving the same variable to give results up to degree three
• solve first-degree equations with non fractional (Applied only) coefficients, using a variety of tools and strategies

•  Just the handouts (see below)

1. Pair students up
2. Have students decide who will be Student A or Student B, and have them complete Problem Set A or B.
3. The answers in each row should match. If they do not match, work together to determine the correct answer.

• See the files in one folder here
• Proportions (Word, PDF)
• Proportions Review (Word, PDF)
• Simplifying Expressions (Word, PDF)
• Adding Polynomials (Word, PDF)
• Simplifying Expressions with Multiplication (Word, PDF)
• Solving Equations (Word, PDF)
• Solving Multistep Equations (Word, PDF)

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

Simplifying Expressions and Solving Equations Tower Challenge

This is a review activity on simplifying expressions and solving equations for grade 9 applied where students answer questions and are rewarded with building materials for each correct answer. The building materials (spagetti & marshmallows) are then used with the goal being the creation of tallest tower. This is based originally on a TIPS activity on quadratics for MBF3C (Unit 3, Day 6).  We have a similar activity for grade 9 academic that can be found here.

MPM 1P

• substitute into and evaluate algebraic expressions involving exponents 
• describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three [i.e., length, which is one dimensional, can be represented by x; area, which is two dimensional, can be represented by (x)(x) or x²; volume, which is three dimensional,can be represented by (x)(x)(x), (x²)(x),or x³]
• add and subtract polynomials involving the same variable up to degree three using a variety of tools
• multiply a polynomial by a monomial involving the same variable to give results up to degree three using a variety of tools
• solve first-degree equations with nonfractional coefficients, using a variety of tools
• substitute into algebraic equations and solve for one variable in the first degree

• 1 bag of spaghetti and 1-2 bags of small marshmallows (or 1 box of straws and 1-inch pieces of tape)  
• a question sheet for each student
• a teacher answer sheet 
• Optional - a whiteboard for each student to work out their solutions
• Optional - prize for the group with the tallest tower

1. Place students in groups (ideally no bigger than 3 per group)
2. Hand out question sheets (and optional whiteboards) to each student.
3. Have students answer questions from their sheet in any order they want. For every correct answer they will get some building materials (eg: 2 spagetti & 3 marshmallows, the amount of each reward is indicated on the student question sheet ). The harder the question the more materials they will get. Eventually the building materials will be used to create a tower with the goal to create the tallest free standing tower.
4. Students work in groups to answer the questions and bring their solutions up to you to be checked. Only one member from each group can come up at a time. Each group can answer each question only once. To keep track of this, use the teacher answer sheet to check off which questions each group has answered as they come up.
5. Leave about 20 min at the end of the class for students to create their towers (students can no longer answer questions)
6. Take lots of pictures and celebrate the group with the tallest free standing tower.

• Gr9AppliedSimplifyingExpression&SolvingEquationsTowerChallengeQuestions (pdf, doc)
• Gr9AppliedSimplifyingExpression&SolvingEquationTowerChallengeTeacherAnswerSsheet (pdf, doc)

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