Fractions That Stop, Fractions That Repeat

We are learning to move both ways between fractions and decimals, including turning a recurring decimal back into an exact fraction, and to use that to define a rational number.

Before you startAbout 50 minutes · what success looks like · notes for the helper

About 50 minutes. It's fine to stop after the Quick Check and come back another day: the lesson opens where you left it, at the Quick Check.

You'll know you've got it when

I can convert a fraction to a decimal and say, with a reason, whether it terminates or recurs
I can convert a recurring decimal to a fraction in simplest form by choosing a multiplier that lines up the repeating block and subtracting
I can define a rational number and explain why every integer, terminating decimal and recurring decimal is one

For the helperhow to sit beside this lesson, and what this step is for

This step: Spark

Start with the question at the top and chat about it. There is no wrong answer here: it gets your learner thinking about what they already know.

Getting ready: Paper, a pencil and a calculator if you have one. A sheet with a few fractions already converted last year (1/3, 1/8, 5/6) can be a useful warm start. The 0.999… Debate needs you or another family member as the other side. At 50 minutes this usually fits one sitting.
What understanding sounds like: Listen for your young person predicting before dividing ('12 has a factor of 3, so it will repeat'), for an explanation of WHY the subtraction works ('multiplying by 100 shifts the decimal two places, so the repeating parts line up and cancel'), and for a definition of rational that is about the fraction, not the decimal: 'it's rational because I can write it as 5/6'. A reason shows more than a correct answer alone.
A likely mix-up, and something that helps: The likely slip is writing 0.777… as 7/10, treating the recurring decimal as if it stopped after one digit. You might ask, 'Is 7/10 the same as 0.7000…? Where did the other sevens go?'
If it comes easily: For a challenge, you might ask whether 0.999… is less than 1, and let them settle it with the same subtraction method. It is a famous argument and worth a few minutes of debate.
If it's a struggle: If it is a struggle, one way is to stay with single-digit repeats (0.4̇, 0.7̇) until the 'multiply by 10, subtract' pattern feels natural, then move to two-digit blocks with the same layout written underneath. Rational, recurring and terminating are words worth saying aloud together, in your home language too if that helps. If they dig in or shut down, one converted decimal checked by dividing is a real result for today.

Step 1 of 6 · Spark

Lesson 1 of 5 · Real Numbers and the Limits of Measuring

Can every decimal go home?

For the helper

Start with the question at the top and chat about it. There is no wrong answer here: it gets your learner thinking about what they already know.

Last year you saw that turning a fraction into a decimal always ends one of two ways: 3/8 stops at 0.375, and 5/6 settles into 0.8333… forever. Today's question runs the other way. If someone hands you 0.272727… or 0.1666…, can you always find the fraction it came from? And here is a harder one to sit with: is 0.999…, with nines going on forever, a little bit less than 1, or is it exactly 1?

0.272727… → ? / ? A repeating decimal is a fraction in disguise. The job today is to take the disguise off.