Grade 7 Number System: negative numbers and fractions, and what to check

What it is

The state describes the domain as applying and extending earlier understanding of fractions to add, subtract, multiply and divide rational numbers. Grade 7 adds negative numbers to everything a child already did with fractions and decimals:

• Adding rational numbers on a number line, and seeing that a number and its opposite add to 0 (NY-7.NS.1).
• Subtracting as adding the opposite, so p − q = p + (−q), and distance on the number line as an absolute value (NY-7.NS.1c).
• Multiplying and dividing signed numbers, and why a negative times a negative is positive (NY-7.NS.2).
• Solving real-world problems that use all four operations (NY-7.NS.3).

Five things to check

1. Number line: starting at −3, a child moves 5 to the right. Where do they land, and can they show it?
2. Subtracting a negative: what is 4 − (−6), and why?
3. Signs in products: −7 × −3 and −7 × 3, said aloud with the rule.
4. Fractions with signs: −1/2 + 1/3. This checks fractions and signs together.
5. A story: a temperature is −4 degrees and drops 7 more. A child who can write the number sentence understands more than one who guesses.

Sample questions, with answers

Each question was checked by two separate programs. The other answers are the ones a child gets from a typical slip.

1. An equation with a missing expression is shown below.

   $(9.2 - 1.6) + (\ \underline{\ ?\ }\ ) = 0$

   Which expression makes the equation true?

   • A. $-9.2 + 1.6$Answer
   • B. $9.2 - 1.6$
   • C. $9.2 + 1.6$
   • D. $-9.2 - 1.6$
   What the other answers come from

   • $9.2 - 1.6$: Same, Not Opposite. The opposite has the other sign. Together they make zero.
   • $9.2 + 1.6$: Second Sign Only. The opposite changes every part, not just the second.
   • $-9.2 - 1.6$: First Sign Only. The opposite changes every part, not just the first.

2. An equation with a missing expression is shown below.

   $(3.6 - 1.3) + (\ \underline{\ ?\ }\ ) = 0$

   Which expression makes the equation true?

   • A. $-3.6 + 1.3$Answer
   • B. $3.6 - 1.3$
   • C. $3.6 + 1.3$
   • D. $-3.6 - 1.3$
   What the other answers come from

   • $3.6 - 1.3$: Same, Not Opposite. The opposite has the other sign. Together they make zero.
   • $3.6 + 1.3$: Second Sign Only. The opposite changes every part, not just the second.
   • $-3.6 - 1.3$: First Sign Only. The opposite changes every part, not just the first.

3. What is the value of the expression $\left(\dfrac{2}{9}\right) \div \left(-\dfrac{2}{11}\right)$?

   • A. $1\frac{2}{9}$
   • B. $-\frac{4}{99}$
   • C. $\frac{4}{99}$
   • D. $-1\frac{2}{9}$Answer
   What the other answers come from

   • $1\frac{2}{9}$: Sign Check Missed. Same signs divide to a positive; different signs divide to a negative.
   • $-\frac{4}{99}$: Times, Not Divide. This one divides. Flip the second fraction, then multiply.
   • $\frac{4}{99}$: Times and Sign Slip. Flip the second fraction and multiply, then check the sign of the answer.

4. What is the value of the expression $\left(-\dfrac{9}{11}\right) \div \left(-\dfrac{3}{4}\right)$?

   • A. $\frac{27}{44}$
   • B. $-\frac{27}{44}$
   • C. $-1\frac{1}{11}$
   • D. $1\frac{1}{11}$Answer
   What the other answers come from

   • $\frac{27}{44}$: Times, Not Divide. This one divides. Flip the second fraction, then multiply.
   • $-\frac{27}{44}$: Times and Sign Slip. Flip the second fraction and multiply, then check the sign of the answer.
   • $-1\frac{1}{11}$: Sign Check Missed. Same signs divide to a positive; different signs divide to a negative.

What a gap looks like

• Ignoring the sign and answering with the right size but the opposite sign.
• Treating subtraction as always making a number smaller, so 4 − (−6) comes out as −2.
• Adding across the top and bottom of fractions: 1/2 + 1/3 = 2/5.

A sign error is usually a rule memorised without a picture. A fraction error usually points to grades 4 and 5. Either way the fix is earlier than grade 7; the plan is in your 7th grader scored Level 1.

Keep reading

• NYS grade 7 math test: format, topics and calculator
• How to find math gaps in a 7th grader