Free ILTS Mathematics (K-6) (407) Study Guide

Domains, Progressions, and Interconnections

The standards group grade-level expectations into domains, the strands running across grades (Counting and Cardinality; Operations and Algebraic Thinking; Number and Operations in Base Ten; Number and Operations, Fractions; Ratios and Proportional Relationships; The Number System). A progression is how one idea grows in sophistication across grades, extended and formalized grade to grade. The standards are interconnected: base-ten place value feeds decimals, fractions feed ratios, and whole-number operation properties are extended, never discarded, as the number system widens.

• Domain vs. standard: a domain is the strand (Base Ten); a standard is a specific grade-level expectation inside it.
• Vertical progression: Counting and Cardinality (K) feeds Base Ten (grades 1-2), then decimal place value (grades 4-5), then the full rational and real system (grade 6 and up).
• Horizontal interconnection: fraction equivalence (grade 4) is the same reasoning that later drives equivalent ratios (grade 6).

What Number Understanding Looks Like at Each Stage

Number skills develop in a predictable order, each benchmark resting on the one before, so the sequence lets you diagnose where a student sits and what comes next.

The Mathematical Practices You Teach Alongside Content

You teach students to do mathematics: to understand and solve problems, to reason and critique the reasoning of others, and to use models and tools appropriately and precisely.

• Understand and solve problems: a student who restates a problem, plans a path, and checks reasonableness is problem solving, not just computing. Ask "what is the problem asking?" before arithmetic.
• Reason and critique reasoning: a second grader who explains why 34 plus 8 crosses into a new ten, and spots a classmate's flaw, is reasoning. Build it with "convince me" and "do you agree, and why?"
• Use models and tools precisely: choosing base-ten blocks for regrouping, a number line for fraction placement, or a double number line for a rate, with correct unit labels.

The Principles Behind Meaningful Counting

Real counting rests on four principles. A child can recite number words and still not truly count until all four are in place.

A kindergarten student who counts five counters, is asked "how many?" and recounts, has one-to-one correspondence and stable order but not cardinality. A student who insists the count "starts over" from a different object lacks order irrelevance.

Language Shapes How Base Ten Is Learned

Early number development is not purely universal; the number-naming structure of a child's home language affects how quickly base-ten structure becomes visible. Some languages name numbers transparently ("ten-one, ten-two" for eleven and twelve), so the tens-and-ones structure is heard directly. English hides it behind irregular words like eleven, twelve, and reversed-sounding teens (fourteen says the ones before the tens).

• Transparent naming exposes structure: when the word states its tens and ones, students map quantity to place value with less friction.
• English irregularities are a known hurdle: the teen words and eleven, twelve must be taught explicitly.
• Honor the resource: a multilingual student's home-language counting is an asset; connect English words to it rather than replace it.

Four Ways to Make Place Value Visible

The place value system gives each digit a value set by its position, with each place worth ten times the place to its right. Reveal that structure with coordinated representations:

• Oral counting: counting by tens and hundreds and bundling ("thirty-four is three tens and four ones") lets students hear the grouping.
• Drawings: quick sticks-and-dots sketches represent tens and ones without full manipulatives.
• Layered place value cards: cards that stack so 200, 30, and 4 overlay to read 234, then pull apart to expose the hidden zeros, linking numeral to expanded value.
• Numerical expressions (expanded form): $234 = 200 + 30 + 4$, later $234 = 2 \times 100 + 3 \times 10 + 4 \times 1$, makes each place explicit in symbols.

Base-ten blocks show 234 as two hundred-flats, three ten-rods, and four unit cubes:

Comparing Integers, Fractions, and Decimals

Order relations ($ \lt $, $=$, $ \gt $) compare numbers by position on the number line: farther right is greater. The logic differs by type:

• Integers: compare place by place from the greatest place; $-3 \lt -1$ because $-3$ sits farther left.
• Fractions: with the same whole, use common denominators, common numerators, or a benchmark such as $\frac{1}{2}$ (Section 5).
• Decimals: align by place value, not digit count. $0.4 \gt 0.39$ because four tenths exceeds three tenths, though $0.39$ shows more digits.

Commutative, Associative, and Distributive Properties

Place value plus three properties power efficient computation. Students compose and decompose numbers to make problems easier, and every standard algorithm is a shortcut these properties justify.

• Commutative: order does not change the result for addition or multiplication. $6 + 9 = 9 + 6$; $4 \times 7 = 7 \times 4$.
• Associative: grouping does not change the result. $(8 + 2) + 5 = 8 + (2 + 5)$; make a ten first.
• Distributive: $a \times (b + c) = a \times b + a \times c$, which breaks a factor apart by place value.

Worked example (distributive): compute $6 \times 27$ by decomposing 27 into $20 + 7$:

Number Lines and Base-Ten Blocks for Decimals

Decimals extend base ten to the right: each place is one tenth of the place to its left, so tenths, hundredths, and thousandths continue the ten-times pattern. Two models make this concrete:

• Number line: the segment 0 to 1 partitioned into ten equal parts shows tenths; partition again for hundredths. Placement makes $0.3 = \frac{3}{10}$ visible.
• Base-ten blocks (redefined whole): let the flat equal one whole; the rod is one tenth and the cube one hundredth. The same blocks that modeled 234 model $2.34$ when the whole is redefined.

A tenths number line placing $0.3$:

Justifying computation: to add $0.3 + 0.24$, line up place values, not last digits. Write $0.30 + 0.24$: three tenths plus two tenths is five tenths, four hundredths gives $0.54$. Verification: $\frac{30}{100} + \frac{24}{100} = \frac{54}{100} = 0.54$. Aligning decimal points is aligning like place values.

Reading the Reasoning Behind Any Algorithm

An algorithm is a reliable step-by-step computing procedure. Interpret standard and student-invented algorithms alike by asking what place-value idea each step relies on. A student-invented method that reaches the right answer through valid place-value reasoning is legitimate, even when its layout differs.

• Standard addition: the "carry" is composing ten ones into one ten, an application of place value.
• Partial sums (invented): $47 + 38 = (40 + 30) + (7 + 8) = 70 + 15 = 85$ exposes regrouping instead of hiding it in a carry.
• Interpreting an error: a student who writes $47 + 38 = 715$ added the tens and ones as if all were ones; the fix is place-value alignment, not "be careful."

Widening the Number System

The number line grows both directions and fills in. Students extend counting numbers to a full system while ordering and operation logic stays intact:

• Negative numbers: the line extends left of zero; $-5$ is the opposite of $5$, and $-5 \lt -2$ because it sits farther left.
• Rational numbers: any number expressible as $\frac{a}{b}$ with integer $a$ and nonzero integer $b$, including terminating and repeating decimals such as $0.75$ and $0.\overline{3}$.
• Irrational numbers: numbers not expressible as such a fraction, with decimals that never terminate or repeat, such as $\pi$ and $\sqrt{2}$. Rationals and irrationals together fill the line as the real numbers.

Misconceptions, Activities, and Assessment

(1) MISCONCEPTIONS TO ADDRESS. Recounting to say "how many" (cardinality not built); reading 34 without knowing the 3 means three tens (digit-as-face-value error); believing a longer decimal is larger, or that zero has no place-holding role in 205.

(2) ACTIVITIES, RESOURCES, TECHNOLOGY. Ten-frames and rekenreks build subitizing and the make-a-ten structure; base-ten blocks and layered place value cards link quantity to numeral; number-line and hundred-chart routines build sequence and magnitude; virtual manipulatives and place-value apps give self-paced practice.

(3) ASSESSMENT FOR DIVERSE NEEDS. A brief counting interview reveals the missing counting principle; a "show me 34 with blocks" task reveals place value; a give-a-number task ("hand me 7 counters") checks cardinality more sharply than counting a set. Offer oral, drawn, and manipulative response modes so language is not a barrier.

Length, Area, and Discrete Models

A fraction can be represented three ways, and strong understanding uses all three.

The length model, placing $\frac{3}{4}$ between 0 and 1:

The Building Blocks of Fraction Sense

• Unit fraction: numerator 1, such as $\frac{1}{4}$. Every fraction is copies of a unit fraction: $\frac{3}{4}$ is three $\frac{1}{4}$'s. This is the foundation of fraction addition.
• Benchmark fractions: reference points, especially $\frac{1}{2}$, with $0$ and $1$, used to estimate and compare. $\frac{5}{8}$ is a little more than $\frac{1}{2}$; $\frac{3}{8}$ a little less.
• The whole (referent unit): a fraction has meaning only relative to its whole. Half of a large pizza is more food than half of a small one, so equal fractions may name different amounts; comparisons require the same whole.
• Equivalent fractions: different numerals naming the same point and amount, $\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}$, generated by multiplying numerator and denominator by the same nonzero number.

Equivalent fractions as equal-length bars: $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{4}{8}$ shade the same amount of the same whole.

The Same Properties, Now With Fractions

The commutative, associative, and distributive properties carry over from whole numbers to fractions unchanged. Adding fractions is adding copies of unit fractions, so a common denominator makes the units match:

The commutative property still holds: $\frac{2}{3} + \frac{4}{9} = \frac{4}{9} + \frac{2}{3}$. Multiplying fractions distributes just as with whole numbers, so the fraction-multiplication area model mirrors the whole-number one.

A Fraction Is a Division; A Ratio Uses the Same Reasoning

A fraction is a quotient: $\frac{3}{4}$ means $3 \div 4$, the amount each gets when 3 wholes are shared among 4. This ties fractions to division, and equivalent fractions in turn power ratios and rates.

• Fraction as division: $\frac{a}{b} = a \div b$. Sharing 3 sandwiches among 4 children gives each $\frac{3}{4}$ of a sandwich.
• Ratio: a comparison of two quantities, such as 3 cups flour to 2 cups sugar, written $3 : 2$.
• Rate: a ratio of quantities in different units, such as 120 miles in 2 hours.
• Unit rate: a rate per one unit, such as 60 miles per hour, found by dividing: $\frac{120}{2} = 60$.

Solving Ratio and Rate Problems With Models

Equivalent ratios are an application of equivalent fractions: scaling both quantities by the same factor keeps the comparison the same, just as scaling a fraction's parts keeps its value. A ratio $3 : 2$ scales to $6 : 4$, $9 : 6$, $12 : 8$. Four models solve these problems:

• Ratio table: rows of equivalent pairs, scaled by multiplying both entries.
• Tape diagram: bars in the ratio's proportion that scale together.
• Number line and double number line: two aligned lines, one per quantity, with tick marks that stay paired as they scale.

A double number line for the flour-to-sugar ratio $3 : 2$:

Proportional and linear relationships: a proportional relationship graphs as a straight line through the origin, with the unit rate as its slope. Two quantities are inversely proportional when their product is constant, linking to reciprocal reasoning: if $y$ is inversely proportional to $x$, then $y$ varies as $\frac{1}{x}$, so doubling $x$ halves $y$.

Connecting Fractions to Decimals and Ordering a List

Decimal fractions have denominators of 10, 100, 1000, and write directly as decimals: $\frac{3}{10} = 0.3$, $\frac{54}{100} = 0.54$. To compare them, compare place by place, or annex zeros so both show the same places ($0.30$ versus $0.24$). Any fraction converts to a decimal by its division: $\frac{3}{4} = 3 \div 4 = 0.75$.

Strategies to order a list of fractions (justify, do not cross-multiply blindly):

• Common denominators: rename so pieces are the same size, then compare numerators.
• Common numerators: with equal numerators, the larger denominator names the smaller fraction ($\frac{3}{5} \gt \frac{3}{7}$).
• Benchmark fractions: sort by whether each is below, at, or above $\frac{1}{2}$.
• Reasoning about size: $\frac{7}{8}$ is one small piece short of a whole; $\frac{5}{6}$ is one bigger piece short, so $\frac{7}{8} \gt \frac{5}{6}$.

Worked example (order $\frac{3}{4}$, $\frac{5}{8}$, $\frac{2}{3}$ least to greatest): LCD of 4, 8, 3 is 24.

Misconceptions, Activities, and Assessment for Fractions

(1) MISCONCEPTIONS. Treating numerator and denominator as separate whole numbers ("$\frac{1}{8} \gt \frac{1}{4}$ because 8 is bigger"); adding across ($\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$); ignoring the whole when comparing; assuming a longer decimal is larger.

(2) ACTIVITIES, RESOURCES, TECHNOLOGY. Fraction bars and strips for equivalence; pattern blocks and geoboards for area partitioning; number lines for placement; paper folding to generate equivalents; virtual fraction manipulatives and interactive number lines for practice with feedback.

(3) ASSESSMENT FOR DIVERSE LEARNERS. Ask students to place a fraction on a number line and explain, to compare two fractions with a benchmark, or to represent one fraction in length, area, and set models. Provide manipulatives and visual response options so a misconception is not hidden behind computation, and pre-teach vocabulary (numerator, denominator, whole) for multilingual learners.