Use one half as a benchmark to classify and compare fractions.
Teaching companion · Version 2.0.0
Use one half as a benchmark, and know when that benchmark cannot finish a comparison.
The original lesson PDFs remain version 1.0.0. This supplement adds diagnostic probes, teaching responses, and question-specific rubrics. Use the supplement below for the updated guidance.
Ask for an explanation or drawing before choosing a response. A wrong answer alone does not identify its cause.
Without dividing, decide whether 3/7 is below or above one half.
Look for: Below one half: two copies give 6/7, still below one whole.
If you observe: Rounds half of seven to three and calls 3/7 exactly one half.
Respond: Pair two copies of the three selected sevenths; notice that six pieces still leave one seventh unfilled.
Both 3/4 and 5/6 exceed one half. Does that fact tell which is larger?
Look for: No. More evidence is needed; 9/12 < 10/12 shows 3/4 < 5/6.
If you observe: Orders them using only their shared “above half” category.
Respond: Place both in the above-half region, then repartition into twelfths to locate their distinct endpoints.
Check independently: Where does 5/11 lie relative to one half? Explain with twice the numerator.
Below half, because 2 × 5 = 10 is less than 11.
Use the benchmark comparison short intervention if the difficulty persists. Return to this check before repeating the exit ticket.
e1 meets its criterion: e1 gives > for 7/12 compared with 1/2. e2 also meets its criterion: e2 explains that comparing twice the numerator with the denominator compares two copies of the fraction with one whole.
Continue to the linked next lesson. Choose common units or another argument when fractions lie on the same side of one half.
Next lesson: Compare And Defende1 does not yet meet this criterion: e1 gives > for 7/12 compared with 1/2.
Ask the learner to explain the attempt, then use the matching diagnostic probe to identify the difficulty. Work through the focused reteach below and its check. Use the linked short intervention if the difficulty persists; retry e1 and e2 before moving on.
Open the linked short interventione1 meets its criterion, but e2 does not yet show this evidence: e2 explains that comparing twice the numerator with the denominator compares two copies of the fraction with one whole.
Use two seven-twelfth selections and ask the learner to connect 14 > 12 with the original one-half comparison, not just recite “double it.”
Open the linked short interventionUse these as follow-up prompts, then examine the learner’s model or explanation. Open responses may differ from the sample while satisfying the criteria.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Look for in the response:
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Look for in the response:
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Look for in the response:
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Look for in the response:
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Look for in the response:
Student pages and teacher answers are separate. Download an HTML file to keep or use your browser to print or save as PDF.
Eight core questions and a two-question exit ticket. Print one set per student.
Recognize a whole and count equal parts.
Look at the first model or task. What is the whole? What do you notice?
Draw columns below, equal to, and above one half.
Place fraction cards by comparing twice the numerator with the denominator.
Partners explain a tricky card with a model rather than a decimal approximation.
Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Compare 7/12 and 1/2. Choose <, =, or >. The wholes are the same size. How does twice the numerator help?
Judges position relative to one half from numerator or denominator alone.
For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Teacher answer key. Open responses include a rubric and one valid sample.
7/12 > 1/2. Rename them as equal-sized parts or place them on the same unit number line.
3/8 < 1/2. Rename them as equal-sized parts or place them on the same unit number line.
4/8 = 1/2. Rename them as equal-sized parts or place them on the same unit number line.
5/6 > 1/2. Rename them as equal-sized parts or place them on the same unit number line.
2/5 < 1/2. Rename them as equal-sized parts or place them on the same unit number line.
6/10 > 1/2. Rename them as equal-sized parts or place them on the same unit number line.
3/7 < 1/2. Rename them as equal-sized parts or place them on the same unit number line.
5/10 = 1/2. Rename them as equal-sized parts or place them on the same unit number line.
1/4 < 1/2. Rename them as equal-sized parts or place them on the same unit number line.
2/4 = 1/2. Rename them as equal-sized parts or place them on the same unit number line.
3/4 > 1/2. Rename them as equal-sized parts or place them on the same unit number line.
1/3 < 1/2. Rename them as equal-sized parts or place them on the same unit number line.
Half of twelve is six; seven twelfths is above one half.
1/3 and 3/8 are both below half.
It shows both exceed half, but more reasoning is needed; 9/12 is below 10/12.
3/5 is greater than half and less than one.
7/12 > 1/2. Rename them as equal-sized parts or place them on the same unit number line.
Compare two copies of the fraction with one whole; twice the numerator versus the denominator gives the relation.
Use the whole and equal-part prompts before moving to independent practice. Some items revisit the core concept with simpler models.
Explain, transfer, and critique. More than one valid response may satisfy the rubric.
Designed for Grade 4. Use one half as a benchmark to classify and compare fractions.
Version 1.0.0. Original, AI-assisted lesson content. Free local classroom use; public reuse terms have not yet been assigned.
These lessons have not yet been field-tested in classrooms. Use the exit ticket and student explanations to decide what to revisit.