Here we have two sets of cards. On one set we have rational equations that start with simple ratio type questions and move to rational equations with polynomials. The second set are similar in style but are created from rational inequalities. In each case, as students move through the cards, the questions become more difficult.

Note that if you look at the expectations, they both say "simple" equations/inequalities but as you move to the last few cards, you may not consider those simple. That is purposeful. We create the cards so that the first one is easy enough for everyone but the last one(s) may go beyond the curriculum so that kids who are not struggling have some place to go that may be challenging. As a teacher, you should decide which card is the level that you hope all students will reach. For other rational function activities to investigate their nature, check out our other post here.

Typically when we have done these, students could check their answers by using a UV pen to reveal the answers written on the answer cards (see example from our fraction continuum to the right). This adds a bit of "magic" to this activity that the students tend to enjoy making them work just a little harder.

- MHF4U 3.6 - solve simple rational equations in one variable algebraically, and verify solutions using technology
- MHF4U 4.2 - determine solutions to polynomial inequalities in one variable [e.g., solve f(x) ≥ 0, where f(x) = x
^{3}– x^{2}+ 3x – 9] and to simple rational inequalities in one variable by graphing the corresponding functions, using graphing technology, and identifying intervals for which x satisfies the inequalities

- Enough copies of each of the question cards for your class (there are six cards per page at each level except the last) 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.
- 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 are sample answers at the end of each document. That is for you to carry around (or not) but not for showing students - more for your reference.
- The "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.

- 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 4 questions (you might choose less or more questions to answer but typically choose a number so that if they don't get them all correct, there are enough questions left on the card for them to practice more).
- 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 giving back the exit card write down the colour card they will start in.
- Students may decide to do one question at a time and then go check their answer or they may do all 4 and then check. Students are monitoring themselves so they decide. If they get the first 4 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 2 or just do enough to get a total of four correct.
- As they move through the continuum, the hope is that they reach the level that you decide equates to "simple" but not too hard4. You may wish to have them do all the questions on that card.

- Rational Equations Continuum Google Doc, PDF
- Rational Inequalities Continuum Google Doc, PDF

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